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WORKS    ON  ^ 

DESCRIPTIVE   GEOMETRY, 

AND  ITS  APPLICATIONS  TO 

ENGINEERING,  MECHANICAL  AND  OTHER  INDUSTRIAL  DRAWING. 
By  S.  EDWARD  WARREN,  C.E. 


I.   ELEMENTARY  WORKS. 

Primary  Geometry.  An  introduction  to  geometry  ao 
usually  presented ;  and  designed,  first,  to  facilitate  an  earlier 
beginning  of  the  subject,  and,  second,  to  lead  to  its  graphical 
applications  in  manual  and  other  elementary  schools.  With 
numerous  practical  examples  and  cuts.  Large  12mo,  cloth,  75c. 
Industrial  Science  Series  Comprising: 

1.  Free-hand  Geometrical  Drawing,  widely  and  variously 
useful  iu  training  the  eye  and  hand  in  accurate  sketching  of  i)lane 
and  solid  figures,  lettering,  etc.  12  folding  plates,  many  cuts. 
Large  12mo,  cloth,  $1.00. 

2.  Drafting  Instruments  AND  Operations.  A  full  descrip- 
tion of  drawing  instruments  and  materials,  with  applications  to 
useful  examples  ;  tile  work,  wall  and  arch  faces,  ovals,  etc.  7 
folding  plates,  many  cuts.     Large  12mo,  cloth,  $1.25. 

3.  Elementary  Projection  Drawing.  Fully  explaining,  in 
six  divisions,  the  principles  and  practice  of  elementary  plan  and 
elevation  drawing  of  simple  solids  ;  constructive  details  ;  shadows ; 
isometrical  drawing ;  elements  of  machines ;  simple  structures. 
24  folding  plates,  numerous  cuts.     Large  12mo,  cloth,  11.50. 

This  and  No.  3  are  especially  adapted  to  scientific,  preparatory, 
and  manual-training  industrial  schools  and  classes,  and  to  all 
mechanics  for  self-instruction. 

4.  Elementary  Perspectiye.  With  numerous  practical 
examples,  and  every  step  fully  explained.  Revised  Edition 
(1891).     Numerous  cuts.     Large  12mo,  cloth,  $1.00. 


5.  I'LANE  Problems  on  the  Point,  Straight  Line,  and  Circle. 
225  problems.  Many  on  Tangencies,  and  other  useful  or  curious 
ones.     150  woodcuts,  and  plates.     Large  12mo,  cloth,  $L25. 

n.   HIGHER  WORKS. 

1.  The  Elements  of  Descriptive  Geometry,  Shadows 
AND  Perspective,  with  brief  treatment  of  Trehedrals ;  Trans- 
versals; and  Spherical,  Axonometric,  and  Oblique  Projections;  and 
many  examples  for  practice.    24  folding  plates.    8vo,  cloth,  $3.50. 

2.  Problems,  Theorems,  and  Examples  in  Descriptive 
Geometry..  Entirely  distinct  from  the  last,  with  115  problems, 
embracing  many  useful  constructions ;  52  theorems,  including 
examples  of  the  demonstration  of  geometrical  properties  by  the 
method  of  projections  ;  and  many  examples  for  practice.  24  fold- 
ing plates.     8vo,  cloth,  62.50. 

3.  General  Problems  in  Shades  and  Shadows,  with 
practical  examples,  and  including  every  variety  of  surface.  15 
folding  plates.    8vo,  cloth,  |>3.00. 

4.  General  Problems  in  the  Linear  Perspective  of 
Form,  Shadow,  and  Reflection.  A  complete  treatise  on  the 
jirinciplcs  and  jjractice  of  perspective  by  various  older  and  recent 
methods  ;  in  98  problems,  24  theorems,  and  with  17  large  plates. 
Detailed  contents,  and  numbered  and  titled  topics  in  the  larger 
problems,  facilitate  study  and  class  use.  Revised  edition,  correc- 
tions, changes  and  additions.    17  folding  plates.    Bvo,  cloth,  $3.50. 

5.  Elements  of  Machine  Construction  and  Drawing. 
73  jiractical  examitles  drawn  to  scale  and  of  great  variety  ;  besides 
30  problems  and  31  theorems  relating  to  gearing,  belting,  valve- 
motions,  6crew-])ropellcrs,  etc.  2  vols.,  8vo,  cloth,  one  of  text, 
one  of  34  folding  plates.    $7.50. 

0,  Problems  in  Stone  Cutting.  20  problems,  with  exam- 
jjle.s  for  j)ractice  under  them,  arranged  according  to  dominant 
surface  (plane,  devclojjablc,  warped  or  double-curved)  in  each,  and 
enibraciiig  every  variety  of  structure  ;  gateways,  stairs,  arches, 
dometr,  winding  ])assages,  etc.  Elegantly  printed  at  the  Riverside 
Press.     10  folding  plates.    8vo,  cloth,  $2.50. 


INDUSTRIAL   SCIENCE  DRAWING. 

ELEMENTART 

PROJECTION  DRAWING. 

THEOJRY    AND     PK.ACTIOE. 


KOR  PREPARATORY  AND  HIGHER  SCIENTIFIC  SCHOOLS  ;  INDUSTRIAL,  AND  NORMAL  CLASSES ;  A_ND 
THE  SELF- INSTRUCTION  OF  TEACHERS,  INVENTORS,  DRAFTSMEN,  AND  ARTISANS. 


Sn  Six  Diuisions. 

DIV.     I.  ELEMENTARY  PROJECTIONS. 

DIV.  II.  DETAILS  OF  BIASONRY,  WOOD,  AND  METAL  CONSTKUCTIONS. 

DIV.  ni.  ELEMENTARY  SHADOWS  AND  SHADING. 

DIV  IV.  ISOMETRICAL  AND  OBLIQUE  PROJECTIONS. 

DIV.    V.  ELEMENTS  OF  BIACHINES. 

DIV.  VI.  SIMPLE  STRUCTURES  AND  MACHINES. 


By  S.  EDWAED  WAEEEI^,  C.E., 

FORMER  PROFESSOR  IN  THE  RENSSELAER  POLYTECHNIC  INSTITUTE,  MASS.  INST.  OPTECHNOIOGE 

AND  BOSTON  NORMAL  ART  SCHOOL;  AND  AUTHOR  OF  A  SERIES  OP  TEXT-BOOKS 

IN  DESCRIPTIVK  GEOMETRY  AND    INSTRUMENTAL    DRAWLNQ. 


THIRTEENTH  EDITION. 


FIFTH   THOUSAND, 
NEW    YORK  : 

JOHN   WILEY  AND   SONS, 

53  East  Tenth  Street, 

1893. 


Copyright, 

1880, 

By  JOHN  WILEY  &  SONS. 


raiM  or  I.  I.  iiTTii  ii  CO., 

KOI.    tfi    10   t6    A^TOR    PLACC,    NIW    TMI, 


CONTENTS. 


Note  to  the  Fourth  Edition,          ......  vi 

Note  to  the  Fifth  Edition,    .......  vii 

From  the  Original  Preface, ix 

Preliminary  Notes — Drawing  Instruments  and  Materials,         .  xi 

DIVISION  I.— PROJECTIONS. 

Chapter  I. — First  Principles. 

§  I. — The  purely  geometrical   or  rational  theory  of  pro- 
jections,     ........  1 

§  II. — Of  the  relations  of  lines  to  their  projections,  .  3 

§  III. — Physical  theory  of  projections,       ....  5 

§  IV.  — Conventional  mode  of  representing  the  two  planes  of 
projection,  and  the  two  projections  of  any  object 
upon  one  plane;  viz.,  the  plane  of  the  paper,      .  5 

§  V. — Of  the  conventional  direction  of  the  light ,  and  of  the 

position  and  use  of  heavy  lines,  .         .         ,  6 

§  VI.— Notation, 7 

§  VII. — Of  the  use  of  the  method  of  projections,         .         .  8 

Chapter  II.  — Projection  of  Li7ies.     Problems  in  Eight  Projection  ; 
and    including    Projections   showing  two  Sides  of  a 
Solid  Right  Angle.     (Thirty-two  Problems.) 
§  I. — Projections  of  straight  lines,  ....  9 

§  II. — Right  projections  of  solids, 13 

§  III. — Projections  showing  two  sides  of  a  solid  right  angle     14 

§  IV. — Special  Elementary  Intersections  and  Developments     23 

General  ExamjAes, 31 

DIVISION    II.— DETAILS    OF   MASONRY,    WOOD,    AND 
METAL   CONSTRUCTIONS. 

Chapter  I.  —  Constricctions  in  Masonry. 

§  I. — General  definitions  and  principles  applicable  both  to 

brick  and  stone  work, 33 

§  II. — Brick  work, 33 

§  III.— Stone  work, 36 

A  stone  box-culvert,        ......  37 

2016052 


1"  CONTEXTS. 

PAG  15 

Chapter  II.  —  Constructions  in  Wood. 

1 1. — General  remarks.     (Explanation  of  Scales.)  .         3i) 

§  II. — Pairs  of  timbers  whose  axes  make  angles  of  0°  with 

each  other, 42 

§  III. — Combinations  of  timbers  whose  axes  make  angles  of 

90^  with  each  other, 45 

§  IV. — Miscellaneous  comljinations.     (Dowelling,  &c.)    .         48 
§  V. — Pairs  of  timbers  wliich  are  framed  together  obliquely 

to  each  other, 49 

§  VI. — Combinations  of  timbers  whose  axes  make  angles  of 

180°  with  each  other, 50 

Chapter  III. — Constructions  in  3fetal,  ......         55 

Cage  valve  of  a  locomotive  pump.     Metallic  pack- 
ing for  stuffing-boxes,  &c.,     .         .         .         .         56 

Boiled-iron  beams  and  columns,    ....         60 

DIVISION  in.— ELEMENTARY  SHADOWS  AND 
SHADING. 
CuAPTKU  I. — ShmJmcs. 

§  I. — Facts,  Principles,  and  Preliminary  Problems,        .         66 
§  II. — Practical  Problems.     (Twelve,  with  examples.)     .         70 

Chapter  II. — Shading, 77 

Hexagonal  prism;   cylinder;   cone,    sphere,    and 
model. 

DIVISION  IV.— ISOMETRIC AL  AND  OBLIQUE  PROJEC- 
TIONS. 

Chapter  I.  — First  Principles  of  Jsometrical  Drawing,          .         .  87 

Chapter  II. — Problems  involving  only  Isometric  Lines,           .         .  90 

('n.KPTKH  III. —Problems  invoicing  ]^on-iso7netri^al  Lines,         .         .  95 
Chaptku  IV. — Problems  intohing  the  Construction  and  Equal  Division 

of  Circles  in  Isometrical  Draicing,        ...  99 

Chapter  V.  —  Oblique  Projections, 106 

DIVISION  v.— ELEMENTS  OF  MACHINES. 

Chapter  I.—Principhs.     Supporters  and  Crank  Motions,    .         .       114 
Pillow-block;    standard;    bed    and   guide -rest; 
crank;  ril)bed  eccentric;  grooved  eccentric. 

Chapter  U.  —  Oearing, 126 

Spur-wheel;  bevel  wheels;  screws  and  serpentines; 
worm- wheel. 


CONTENTS. 


DIVISION  VI. -SIMPLE  STRUCTURES  AND  MACHINES. 

Chapter  I. — Stone  Structures,    . 

A  brick  segmental  arch,     . 
A  semi-cyliadrical  culvert,  with  vertical  cylindri- 
cal wing-walls, 
Chapter  II, — Wooden  Structures, 
A  king-post  truss, 
A  queen-post  truss- bridge, 
Chapter  III.  —Iron  Constructions, 

A  railway  track— frog,  chair,  &c., 
A  hydraulic  ram,    . 
Exercises.— A  stop-valve;  a  Whipple  truss-bridge; 
a.  vertical  boiler;  a  Knowles  steam-pump. 


140 
140 

112 
111) 
14G 
147 
151 
151 
153 


NOTE  TO  THE  FOURTH  EDITION. 


This  edition  is  improved  cliicfly  by  the  extension  of  the  chapter 
on  obUque,  or  pictorial  projections,  also  called  mechanical  per- 
ppective,  with  the  addition  of  a  new  plate. 

Numerous  niuior  corrections,  and  new  or  improved  paragraphs 
have  beeti  scattered  through  the  work,  as  suggested  by  further  ex- 
perience. Yet  it  is  not  expected  that  tliis  edition,  however  im- 
proved, can  supersede  either  careful  and  thinking  study  on  the 
pait  of  the  willing  student,  or  ample,  repeated^  and  varied  instruc- 
tion on  tlie  p:irt  of  tlie  willing  teacher.  Many  minds  require 
variously  changed  and  amplified  statements  of  the  same  thing 
before  tht-y  are  ready  to  exclaim  heartily,  "I  see  it  now;"  and  the 
teacher  must  be  ready  to  meet,  with  many  forms  of  instruction,  the 
conditions  presented  to  him  by  different  minds. 

A  very  moderate  collection  of  such  objects  as  are  illustrated  in 
Division  II.,  and  such  as  can  be  made  by  a  carpenter  or  turner,  or 
m  machine,  gas-fitting,  or  pattern  shops,  will  very  usefully  supple- 
ment the  text,  and  add  to  the  pleasure  and  benefit  derived  by  the 
Btudcnt;  and  will  be  better  than  an  increased  size  of  the  book, 
which  is  meant  to  be  rather  suggestive  than  exhaustive,  or  to  be 
too  closely  followed, in  respect  to  the  examples  chosen  for  practice. 
Finally,  the  author's  chief  desire  in  relation  to  this,  as  well  as  the 
rest  of  his  '•  elementaiy  works,"  is,  to  see  them  so  generally  used  in 
hi'jhtr  preparatory  instruction  as  to  give  due  place  to  higher 
•ludies  in  the  same  department  in  the  strictly  Technical  Schools. 

H.  1'.  I.,  TuoY,  July,  1871. 


NOTE  TO  THE  FIFTH  EDITION. 


The  call  for  a  new  edition  of  this  manual  has  led  the  Author, 
after  the  lapse  of  ten  years  since  the  last  revision,  to  make  such 
further  improvements  in  a  new  edition,  as  additional  and  varied 
experience  and  reflection  have  suggested. 

A  few  paragraphs  m  Division  I.  have  been  re-written. 

A  few  but  valuable  additions  have  been  made  to  the  text  and 
plates  of  Div.  II. 

Div.  III.  contains  one  more  plate,  taken  from  Div.  VI.,  and 
examples  of  finished  shading,  not  before  provided. 

Div.  IV.  has  been  improved  by  the  addition  of  a  few  figures 
and  by  partial  re-writing. 

The  most  important  change  is  the  addition  of  Div.  V.,  em- 
bracing the  more  important  and  universal  elements  of  machines. 
The  volume  is  thus  made  more  complete,  both  in  itself,  and  as  an 
introduction  to  higher  works  both  theoretical  and  practical.* 

Div.  VI.  (V.  in  previous  editions)  has  been  slightly  enlarged 
by  a  few  new  and  valuable  practical  examples. 

In  general:  while  the  subjects  of  all  the  Author's  "  Elementary 
Works  "  have  been  largely  taught  in  the  earlier  classes  of  Poly- 
technic Schools,  of  whatever  name,  it  is  to  be  hoped  that  by  the 
increasing  development  of  scientific  instruction,  they  will  all 
ultimately  be  included  in  Preparatory  Scientific  Instruction, 
and  in  Special  Normal  Classes;  and  in  behalf  of  the  many  pupils 
whose  education  ends  in  preparatory  schools,  but  to  whom  an 
exact  knowledge  of  elementary  instrumental  or  mathematical 
drawing  would  be  highly  useful. 

The  explanations  of  first  principles  have  purposely  been  made 
very  complete,  in  behalf  of  all  classes  of  self-instructors,  and 
because  what  is  not  thus  printed  must  be  said,  and  often  repeated, 

*  The  beautiful  plates  XVII.,  XVIII.,  XIX.,  modified  however  to  suit  a 
full  explanatory  text,  are  from  the  Cours  de  dessln  lineaire,  par  Delaistke,  a 
work  which  every  draftsman  would  do  well  to  possess. 


viii  XOTE   TO   THE    FIFTH    EDITION". 

It)  ensure  that  full  understanding  of  the  subject,  the  test  of  which 
is  the  ready  performance  of  new  examples. 

At  this  point  the  testimony  of  an  evidently  experienced  and 
faithful  English  author  and  teacher  may  well  be  noted.  Speak- 
ing of  the  copying  system,  he  says,  "If,  however,  at  the  end  of 
one  or  two  years'  practice,  the  copyist  [though  able  to  make  a 
highly  finished  copy]  is  asked  to  make  a  side  and  end  elevation 
and  longitudinal  section  of  his  lead-pencil,  or  a  transverse  section 
of  his  instrument-box,  the  chances  are  that  he  can  do  neither 
the  one  nor  the  other.  Strange  as  it  may  appear,  this  is  a  state 
of  things  which  I  have  had  frequent  opportunities  for  witness- 
ing. .  .  .  The  remedy  has  been  to  commence  a  course  of 
study  from  the  very  beginning.  ...  He  has  made  from  the 
copy  a  highly  finished  drawing,  with  all  the  shadows  admirably 
projected,  being  at  the  same  time,  however,  perfectly  ignorant 
of  the  rules  for  projecting  such  shadows.  This  is  the  true 
picture  of  a  student  who  had  a  course  of  two  years'  study  where 
mechanical  drawing  was  taught  "  [by  merely  copying  successively 
more  and  more  elaborate  drawings].* 

With  these  remarks  the  present  edition,  in  its  final  form  as 
now  intended,  is  committed  to  the  favor  of  Schools  and  Self- 
Instructors. 

Newton,  Mass.,  October,  1880. 

*  Preface  to  Bln'n's  Orthographic  Projections.    London,  1867. 


FEOM  THE  OPtlGINAL  PEEP  ACE. 


Experience  in  teaching  shows  that  correct  conceptions  of  the 
forms  of  objects  having  three  dimensions,  are  obtained  with 
considerable  difficulty  by  the  beginner,  from  drawings  having 
but  two  dimensions,  especially  when  those  drawings  are  neither 
"  natural " — that  is  "  pictorial " — nor  shaded,  so  as  to  suggest 
their  form  ;  but  are  artificial,  or  "conventional,"  and  are  merely 
**  skeleton,"  or  unshaded,  line  drawings.  Ilence  moderate  experi 
enc-e  suggests,  and  continued  experience  confirms,  the  propriety 
of  interposing,  between  the  easily  understood  drawings  of  pro 
blems  involving  two  dimensions,  and  the  general  course  of  pro- 
blems of  three  dimensions,  a  rudimentary  course  upon  the  methoda 
of  representing  objects  having  three  dimensions. 

Experience  again  proves,  in  respect  to  the  drawing  of  any 
engineering  structures  that  are  worth  drawing,  that  it  is  a  great 
advantage  to  the  draftsman  to  have — 1st,  some  knowledge  of  the 
thing  to  he  drawn^  aside  from  his  knowledge  of  the  methods  of 
drawing  it ;  and  2d,  practice  in  the  leisurely  study  of  the  graphical 
construction  of  single  members  or  elements  of  a  piece  of  framing 
or  other  structure. 

The  truth  of  the  second  of  the  preceding  remarks,  is  further 
apparent,  from  the  fact  that  in  entering  at  once  upon  the  draw- 
ing of  whole  structures,  three  evils  ensue,  viz. — 1st,  Confusion 
of  ideas,  arising  from  the  mass  of  new  objects  (the  many  different 
parts  of  a  structure)  thrown  upon  the  mind  at  once ;  2d,  Losi 
of  time,  owing  to  repetitirn  of  the  same  detail  many  times  \v 


PREFACH. 


the  same  structure  ;  and  3d,  Waste  of  drawings,  as  well  as  of 
time,  through  poor  execution,  which  is  due  to  insufficient  pre 
vious  practice.  Hence  Divisions  II.  and  Y.  contain  a  liberal 
collection  of  elements  of  structures  and  machines,  each  one  of 
which  affords  a  useful  problem,  while  Division  YI.  includes 
examples  of  a  few  simple  structures,  to  fulfil  the  threefold  pur- 
pose of  affording  occasion  for  learning  the  names  of  parts  of 
structures ;  for  practice  in  the  combination  of  details  into  whole 
structures ;  and  for  profitable  review  practice  in  execution. 

Classes  will  generally  be  found  to  take  a  lively  interest  in 
the  subjects  of  this  volume — because  of  their  freshness  to  most 
learners,  as  new  subjects  of  interesting  study — because  of  the 
variety  and  brevity  of  the  topics — and  because  of  the  compact- 
ness and  beauty  of  the  volume  which  is  formed  by  binding  to- 
gether all  the  plates  of  the  course,  when  they  are  well  executed. 
As  to  the  use  of  this  volume,  it  is  intended  that  there  should 
be  formal  interrogations  upon  the  problems  in  the  1st,  3d,  and 
4th  divisions,  with  graphical  constructions  of  a  selection  of  the 
same  or  similar  ones;  and  occasional  interrogations  mingled 
with  the  graphical  constructions  of  the  practical  problems  of 
the  remainini^  divisions.  Ilememberino:  that  excellence  in  mere 
execution,  though  highly  desirable  and  to  be  encouraged,  is 
not,  at  this  stage  of  the  student's  progress,  the  sole  end  to  be 
attained,  the  student  may,  in  place  of  a  tedious  course  of 
finished  drawings,  be  called  on  frequently  to  describe,  by  the 
aid  of  pencil  or  blackboard  sketches,  hoto  he  would  construct 
drawings  of  certain  objects — cither  those  given  in  the  several 
Divisions  of  this  volume,  or  other  similar  ones  proposed  by  his 
teacher. 


PRELIMINARY  NOTES. 


As  beginners  not  seldom  find  peculiar  difficulties  at  the  outset 
of  the  study  of  projections,  the  removal  of  which,  however, 
makes  subsequent  progress  easy,  the  following  special  explana- 
tions are  here  prefixed. 

I.  Figs.  1,  2,  3,  5,  and  15,  of  PI.  I.  are  pictorial  diagrams, 
used  for  illustration  in  place  ofactital  models.  Thus,  in  Fig.  3,  for 
example,  MHi  represents  a  horizontal  square  cornered  plane  sur- 
face, as  a  floor.  MGY  represents  a  vertical  square  cornered 
plane  surface,  as  a  wall,  which  is  therefore  perpendicular  to 
MH^.  P  represents  any  point  in  the  angular  space  included  by 
these  two  planes.  Pj:;  represents  a  line  from  P,  perpendicular  to 
the  plane  MH<,  and  meeting  it  at^.  Vp'  represents  a  line  from 
P,  perpendicular  to  the  plane  AIGV,  and  meeting  it  at^'.  Then 
Vp  and  Vp'  are  called  the  projecting  lines  of  P.  The  point  p  is 
called  the  horizontal  projection  of  P,  and  p'  the  vertical  projection 
of  P.  The  projecting  lines  of  any  point  or  of  any  hody,  as  in 
Fig.  1,  are  perpendicular  to  the  planes,  as  MH^  and  MGV, 
which  are  called  planes  of  projection. 

II.  In  preparing  a  lesson  from  this  work,  the  object  of  the 
student  is,  by  no  means,  to  commit  to  memory  the  figures,  but 
to  learn,  from  the  first  principles,  and  subsequent  explanations, 
to  see  in  these  figures  the  realities  in  space  which  tliey  represent;  so 
as  to  be  able,  on  hearing  the  enunciation  of  any  of  the  problems, 
to  solve  it  from  a  clear  understanding  of  the  subject,  and  not 
"by  rote"  from  mere  memory  of  the  diagrams.  The  student 
will  be  greatly  aided  in  so  preparing  his  lessons,  by  working 
out  the  problems,  in  space,  on  actual  planes  at  right  angles  to 
each  other,  as  on  the  leaves  of  a  folding  slate,  when  one  slate  is 
placed  horizontally  and  the  other  vertically.  In  the  construction 
oi\c\& plates,  he  should  also  test  his  knowledge  oi \\iq. principles, \)'^ 
varying  the  form  of  the  examples,  though  without  essentially 
changing  their  character. 


Xll 


PRELIMINARY   NOTES. 


Drawing  Instruinents  and  Materials., 

This  volume  is  intended  to  be  the  immediate  successor  of 
my  "  Drafting  Instruments  and  Operations,"  which  is  there- 
fore supposed  to  have  been  read  first,'  by  students  of  tliis  one. 

But  as  some  self-instructors  and  other  students  may  desire  to 
acquire  a  knowledge  of  projections  as  quickly  as  possible,  for 
j^ractical  use,  and  without  regard,  at  first,  to  finished  execution 
of  their  drawings,  the  following  condensed  information  is  here 
inserted  for  their  convenience. 

To  abridge  the  descriptions  to  the  utmost,  it  may  first  be 
stated  that  dealers  in  Drawing  Instruments  and  Materials  are 
found  in  all  large  cities,  who  will  send  descriptive  catalogues 
on  a])plication.  Such  are  Frost  &  Adams,  Boston ;  W.  & 
L.  E.  Gurley,  Troy,  N.  Y.;  Keuffel  &  Esser,  New  York  City; 
James  "W.  Queen,  Philadelphia;  and  others,  doubtless,  whose 
advertisements  can  be  found  in  educational  and  popular  me- 
chanical periodicals.     The  necessary  articles  are : — 

1.  A  good^a27'  of  cotnpasscs,  with  tliQiv  furniture;  that  is, 
a  pen,  pencil,  and  needle  point  to  replace  the  movable  steel 
points,  when  drawing  circles  in  pencil  or  ink. 


2.  A  good  drawing  pen. 


o.  A  (b-awing  board  20  x  30  inches. 

4.  A  T  square ;  that  is,  a  hard-wood  rnler,  having  a  stout 
M'oodcn  cross-piece  about  2|-x  9  inches, 
and  half  an  inch  thick,  at  one  end, 
njion  the  fiat  side  of  which  the  blade 
is  firmly  fastened,  truly  at  right  angles.  The  blade  may  be 
about  30  inches  long. 


n 


PKELIMINART    NOTES. 


XIU 


5.  A  pair  of  hard-wood  riglit-angled  triangles,  the  longest 
side  about  10  inches  long ;  one  with  the  two  acute  angles  of 
45°  each,  the  other  with  acute  angles  of  30°  and  60°, 


6.  A  triangular  scale,  graduated  into  tenths  or  twelfths  of 
the  unit  as  may  be  preferred ;  or,  a  flat  ivorj  "  protractor 
scale." 


7.  Buff  manilla  office,  or  "detail"  paper,  or,  if  preferred  and 
it  can  be  afforded,  Whatman's  rough  drawing  paper,  of  con- 
venient size,  from  "medium,"  17"  x  22",  to  "imperial," 
21"  X  30". 

8.  Hard  lead-pencils. 

9.  A  cake  of  Indian  ink — Chinese  the  best  for  shadina:, 
the  Japanese  for  Imes. 

Where  the  utmost  economy  is  sought,  a  very  cheap,  fair 
quality  of  brass  instruments  can  be  had  in  boxes,  or  the  draw- 
ings can  even  be  made  with  pencil  only ;  any  neat  worker  in 
hard  wood  can  make  the  drawing  board,  T  square,  and  triangles, 
and  a  foot-rule  may  be  made  to  serve  as  a  scale. 

When  drawings  are  not  to  be  colored,  the  paper  can  be  lightly 
gummed  or  tacked  to  the  drawing  board  at  the  corners.  Other- 
wise, the  sheet  should  be  well  wet  by  sponging  with  clean 
water  and,  while  wet,  fastened  to  the  board  by  means  of  thick 
mucilage  applied  along  the  edge  of  the  paper. 

Indian  ink  is  prepared  for  use  by  rubbing  it  on  a  bit  of 
china,  with  a  few  drops  of  water.  It  is  then  applied  between 
the  blades  of  the  drawing  pen  by  a  small  feather  or  slip  of 
paper.     Pen  and  ink  should  be  wiped  dry  when  done  with. 


ELEMENTARY   PROJECTION    DRAWING. 


DIVISION  FIRST. 

PROJECTIONS. 


CHAPTER  I. 

FIRST  PRINCIPLES. 

§  1 .     Tlic  purely  Geometrical  or  Rational  Theory  of  Projections. 

1.  Elementary  Projection  Drawing  is  an  introduction  to 
Descriptive  Geometry,  and  shows  how  to  represent  simple  solids, 
singly  and  in  combination,  upon  plane  surfaces,  yet  so  as  to  show 
their  real  dimensions. 

2.  If  ten  feet  of  5-inch  stove-pij)e  were  wanted,  a  circle  five 
inches  in  diameter,  drawn  on  paper,  would  be  all  the  pattern  the 
workman  would  need.  But  if  the  desired  length  were  forgotten, 
or  if  the  pipe  were  to  be  conical,  the  circular  drawing  would  no 
longer  be  sufficient.  That  is,  as  a  plane  surface  has  but  two  di- 
mensions, no  more  than  two  dimensions  of  any  object  can  be  ex- 
actly shown  in  one  figure  on  that  plane. 

But  practical  toork,  and  geometrical  problems  for  study,  are 
both  continually  arising,  which  require,  for  conveniejit  execution 
in  one  case,  and  proper  solution  in  the  other,  that  we  should  be 
able,  in  some  way,  to  truly  show  all  the  dimensions  of  solid 
bodies  upon  plane  surfaces. 

3.  What,  then,  is  the  number  and  the  relative  position  of  the 
planes  which  will  enable  us  to  rejjresent  all  the  dimensions  of  any 
geometrical  solid,  in  their  real  size,  on  those  planes?  To  assist  in  an- 
swering this  question,  reference  maybe  made  to  PL  I.,  Fig.  1.  Let 
ABCFED  be  a  regular  square-cornered  block,  whose  length  is  AB; 
breadth,  AD  ;  and  thickness,  AC  ;  and  let  MN  be  any  horizontal 
plane  below  it  and  parallel  to  its  top  surface  ABED.  If  now  from 
the  four  points  A,  B,  D  and  E,  perpendiculars  be  let  fall  r.pon  the 
horizontal  plane  MN,  they  will  meet  it  in  the  points  a,  b,  d  and  e. 


2  FIRST    PRINCIPLES. 

Bv  joining  these  points,  it  is  evident  that  a  figure — abed — will  be 
formed,  avIucIi  a\i11  be  equal  to  the  top  surface  of  the  block,  and 
will  be  a  correct  representation  of  the  lerigth  and  breadth  of  that 
top  surface — i.  e.  of  the  length  and  breadth  of  the  block.  Simi- 
larly, if  MP  be  a  vertical  plane,  parallel  to  the  front,  ABCF,  of 
the  block;  and  if  perpendiculars,  Aa',  etc.,  be  let  fall  from 
A,  B,  C,  F,  upon  MP;  the  figure,  a'b'c'f,  will  be  equal  to  ABCF, 
and  hence  will  show  the  length  and  thickness  of  the  block. 

4.  From  the  last  article  the  following  definitions  arise.  The 
point  a,  PI.  I.,  Fig.  1,  is  where  A  would  arrive  if  thrown,  that  is, 
projected,  vertically  doAvnwards  along  the  line  Aa.  Likewise,  a' 
is  where  A  would  be,  if  thrown  or  projected  along  Aa',  from  A 
to  a',  perpendicularly  to  the  plane  MP.  Hence  a  is  called  a 
horizontal  projection  of  A  ;  and  a'  is  called  its  vertical  projection. 

Also,  conversely,  A  is  said  to  be  horizontally  i^rojected  at  a, 
and  vertically  projected  at  a'. 

The  plane  MN  is  thence  called  the  horizontal  plane  of  projec- 
tion ;  and  the  plane  MP,  the  vertical  plane  of  pi'ojection. 

Aa  and  Aa'  are  called  the  two  projecting  lines  of  the  point  A 
relative  to  the  planes  of  projection,  MN  and  MP. 

Hence  we  have  this  definition.  If  from  any  given  point  a  per- 
pendicular be  let  fall  upon  a  plane,  the  point  where  that  perpen- 
dicular meets  the  plane  will  be  the  projectio7i  of  the  point  upon  the 
plane,  and  the  perpendicular  will  be  the  projecting  line  of  the  point. 

5.  Similar  remarks  apply  to  any  number  of  points  or  to  objects 
limited  by  such  points;  and  to  their  projections  upon  any  other 
planes  of  projection.  Thus  abed  is  the  horizontal  projection  of 
the  block  ABCD,  and  a'b'c'f  is  its  vertical  projection,  and, 
generally,  the  projecting  lines  of  objects  are  perpendicular  to  the 
planes  of  projection  employed.  Finally,  the  intersection,  as  MR, 
of  a  horizontal,  and  a  vertical  plane  of  projection,  is  the  ground 
line  for  that  vertical  plane. 

6.  From  the  foregoing  articles  the  following  principles  arise. 
First:  Two  planes,  at  right  angles  to  each  other,  are  necessary 
to  enable  us  to  represent,  fully,  the  three  dimensions  of  a  solid. 
Second:  In  order  that  those  dimensions  shall  be  seen  in  their  true 
size  and  relative  jjo.sition,  they  must  be  parallel  to  that  plane  on 
which  they  arc  shown.     Third:  Each  plane  shows  two  of  the  di- 


FIRST   PRINCIPLES.  3 

mcnsions  of  the  solid,  viz.,  the  two  which  are  panillel  to  it;  and 
that  dimension  which  is  thus  shown  twice,  is  the  one  which  is 
parallel  to  both  of  the  planes.  Thus  AB,  the  length,  and  AD, 
the  breadth,  are  shown  on  the  plane  MN  ;  and  AB,  the  length 
again,  and  AC,  the  thickness,  are  shown  on  the  plane  MP. 
Fourth :  The  height  of  the  vertical  projection  of  a  point  above 
the  ground  line,  is  equal  to  the  height  of  the  point  itself,  in 
space,  above  the  horizontal  plane;  and  the  perpendicular  distance 
of  the  horizontal  projection  of  a  point  from  the  ground  line,  is 
equal  to  the  perpendicular  distance  of  the  point  itself,  in  front 
of  the  vertical  plane.  Thus  :  PI.  I.,  Fig.  1,  aa"  =  Aa'  and 
a'a"  =  Aa. 

7.  The  preceding  principles  and  definitions  are  the  foundation  ot 
the  subject  of  projections,  but,  by  attending  carefully  to  Pi.  I.,  Fig. 
2,  some  useful  elementary  applications  of  them  may  be  discovered, 
which  are  frequently  applied  in  practice.  PI.  I.,  Fig.  2,  is  a  pic- 
torial model  of  a  pyramid,  Ycdeg,  and  of  its  two  projections.  The 
face,  Ycd,  of  the  pyramid,  is  parallel  to  the  vertical  plane,  and  the 
triangle,  X«5,  is  equal  and  parallel  to  Ycd,  and  a  little  in  front  and 
at  one  side  of  it.  By  first  conceiving,  now,  of  the  actual  models, 
which  are,  perhnps,  represented  as  clearly  as  they  can  be  by  mere 
diagrams,  in  PL  I.,  Figs.  1  and  2 ;  and  then  by  attentive  study  of 
those  figures,  the  next  two  articles  may  be  easily  imderstood. 

§  II. —  Of  the  Melations  of  Lines  to  their  Projections. 

8.  Relations  of  single  lines  to  their  projections. 

a.  A  vertical  line,  as  AC,  PI.  I.,  Fig.  1,  has,  for  its  horizontal 
projection,  a  point,  a,  and  for  its  vertical  projection,  a  line  a'c', 
perpendicvxlar  to  the  ground  line,  and  equal  and  parallel  to  the  line 
AC,  in  space. 

b.  A  horizontal  lijie,  as  AD,  which  is  perpendicidar  to  the  verti- 
cal plane,  has,  for  its  horizontal  projection,  a  line,  ad,  perpeU' 
dicular  to  the  ground  line,  and  equal  and  parallel  to  the  line,  AD, 
in  space  ;  and  for  its  vertical  projection  a  point,  a  . 

c.  A  horizontal  line,  as  AB,  which  is  parcdld  to  both  planes  of 
projection,  has,  for  both  of  its  projections,  lines  ab  and  a'b',  which 
are  parallel  to  the  ground  line,  and  equal  and  parallel  to  the  line, 
AB,  in  space. 

d.  A  horizontal  line,  as  BD,  which  makes  an  acute  angle  icith 
the  vertical  plane,  has,  for  its  horizontal  projection,  a  line,  bdy 
v^hich  makes  the  same  angle  with  the  ground  line  that  the  line,  BD, 


4  FIRST  PRINCIPLES. 

makes  willi  the  vertical  plane,  and  is  equal  and  parallel  to  the  line 
itself  (BD) ;  and  has  for  its  vertical  projection  a  line  h'a\  which  is 
parallel  to  the  ground  line,  but  shorter  than  BD,  the  line  in  space. 

e.  An  oblique  line,  as  BC,  PI.  I.,  Fig.  1,  or  \d,  I'l,  I.,  Fig.  2< 
which  is  parallel  to  the  vertical  plane,  has,  for  its  vertical  projection, 
a  line  he,  or  v'd\  which  is  equal  and  parallel  to  itself,  and  for  its 
horizontal  projection,  a  line  ha  or  vd,  parallel  to  the  ground  line, 
but  shorter  than  the  line  in  s])ace. 

f.  Au  oblique  line,  as  V^,  PI.  I.,  Fig.  2,  ichich  is  oblique  to  both 
planes  of  projection,  has  both  of  its  projections,  v'd'  and  vg,  oblique 
to  the  ground  line,  and  shorter  than  the  line  itself. 

</.  An  oblique  line,  as  AH,  PI.  I.,  Fig.  1,  which  is  oblique  to  both 
planes  of  projection,  but  is  in  a  plane  ACDH,  perpendicular  to 
both  of  those  planes,  has  both  of  its  projections,  a'c'  and  ad,  perpen- 
dicular to  the  ground  line,  and  shorter  than  the  line  itself. 

h.  A  line,  lying  in  either  plane  of  projection,  coincides  with  its 
projection  on  that  plane,  and  has  its  other  projection  in  the  ground 
line.     See  cd — c'd',  the  projections  of  ccZ,  PI.  I.,  Fig.  2. 

9.  Remarh.  A  general  principle,  which  it  is  important  to  be 
perfectly  familiar  with,  is  embodied  in  several  of  the  preceding 
examples;  viz.  When  any  line  is  j^af^^H'^l  to  either  plane  of  projec- 
tion, its  projection  on  that  plane  is  equal  and  parallel  to  itself,  and 
its  projection  on  the  other  plane  is  parallel  to  the  ground  line. 

10.  The  preceding  remark  serves  to  show  how  to  find  the  true 
length  of  a  line,  when  its  projections  are  given.  When  the  line,  as 
V^,  PI.  I.,  Fig.  2,  is  obli(}ue  to  both  jdanes  of  projection,  its  length, 
V«7,  is  evidently  equal  to  \\\(ihypothenxise  of  a  right-angled  triangle, 
of  which  the  base  is  vg,  the  horizontal  projection  of  the  line,  and  the 
altitude  is  V^',  the  height  of  the  upper  extremity,  Y,  above  the 
horizontal  plane.  When  the  line,  as  All,  PI.  I.,  Fig.  1,  does  not 
touch  either  plane  of  ])rojection,  it  is  evidently  equal  to  the  hypo- 
tlienuse  of  a  right-angled  triangle,  of  Avhich  the  base,  CII,  equals 
the  horizontal  2)rojection,  ad,  and  the  altitude,  AC,  equals  the 
difference  of  the  jterpcjidictdars,  Aa  and  Hr?,  to  the  horizontal  plane 

In  the  same  way,  it  is  also  true  that  the  line,  as  V^,  PI.  I.,  Fig.  2, 
is  the  hypothenuse  of  another  right-angled  triangle,  whose  base 
equals  the  vertical  projection,  v'd',  and  whose  altitude  equals  the 
difference  of  the  perpendiculars,  Vu'  and  d'g,  from  the  extremities 
of  the  line  to  the  vortical  ])lane  of  projection. 

11.  Relations  iA'  pairs  of  lines  to  their  projections.  These  rela- 
tions, after  the  full  notice  now  given  of  the  various  positions  of 
single  lines,  may  be  briefly  expressed  as  follows. 


FIRST   PRINCIPLES.  fi 

a.  A  pair  of  lines  which  are  equal  and  parallel  in  apace.,  andalsa 
parallel  to  a  plane  of  projection,  as  AB  and  CF,  PI.  I.,  Fig.  1,  or  Vc 
and  Xa,  PI.  I.,  Fig.  2,  have  their  projections  on  tliat  plane — a'b' 
and  c'/\  PL  I.,  Fig.  1,  or  v'c'  and  x'a\  PI.  I.,  Fig.  2 — equal  and 
varallel — to  each  other.,  and  to  the  lines  in  space. 

b.  A  pair  of  lines  Avhich  are  equal  and  parallel  in  space,  hut  not 
parallel  to  ap>lane  of  projection,  will  have  their  projections  on  that 
plane  equal  and  parallel  to  each  other,  but  not  to  the  lines  in 
space, 

c.  Parallel  lines  make  equal  angles  with  either  plane  of  projec- 
tion ;  hence  it  is  easy  to  see  that  lines  not  parallel  to  each  other — 
as  Yd  and  Vc,  or  Yg  and  Ve,  PI.  I.,  Fig.  2 — but  which  make 
equal  angles  with  the  planes  of  projection,  will  have  equal  projec- 
tions on  both  planes — i.e.  v'd'  =.v'c'  and  vg—ve,  also  vd=vc. 

§  III. — Physical  Theory  of  Projections. 

12.  The  preceding  articles  comprise  the  substance  of  the  purely 
geometrical  or  rational  theory  of  projections,  which,  strictly,  ia 
sufficient ;  but  it  is  natural  to  take  account  of  the  physical  fact  that 
the  magnitudes  in  space  and  their  representations,  both  address 
themselves  to  the  eye,  and  to  inquire  from  lohat  distance  and  in 
what  direction  the  magnitudes  in  PL  I.,  Figs.  1  and  2,  must  be 
viewed,  in  order  that  they  shall  appear  just  as  their  projections 
represent  thera.  Since  the  projecting  lines,  Q,  regarded  as  rays, 
reflected  from  the  block.  Fig.  1,  to  the  eye,  are  parallel,  they  could 
only  meet  the  eye  at  an  infinite  distance  in  front  of  the  vertical 
plane.  Plence  the  vertical  projection  of  an  object  represents  it  as  it 
would  appear  to  the  eye,  situated  at  an  infinite  distance  from  it,  and 
looking  in  a  direction  perpendicular  to  the  vertical  plane  of  pro- 
jection. Likewise,  the*  projecting  lines,  S,  show  that  tlie  horizontal 
projection  of  an  object  represents  it  as  seen  from  an  infinite  distance 
above  it,  and  looking  perpendicularly  down  upon  the  horizontal 
plane.  Thus,  the  projecting  lines  represent  the  direction  of  vision^ 
which  is  perpendicular  to  the  plane  of  projection  considered. 

§  IV. — Conve7itional  Mode  of  representing  the  two  Planes  of  Pro- 
jection, and  the  two  Projections  of  any  Object  upon  one  plant 
— viz.  the  Plane  of  the  Paper. 

13.  In  practice,  a  single  flat  sheet  of  paper  represents  the  two 
planes  of  projection,  and  in  the  following  manner.  The  vertical 
plane,  IMY,  PL  I.,  Fig.  3,  is  supposed  to  revolve  backwaids,  as 


U  FIRST   PRINCIPLES. 

shown  by  the  arcs  rii  and  V^,  till  it  coincides  with  the  horizontal 
plane  produced  at  M  u  t  G.  Ilence,  drawing  a  line  from  right  to 
loft  across  the  paper,  to  represent  the  ground  line,  MG,  all  that 
joart  of  the  pajDcr  above  or  beyond  such  a  line  will  represent  the 
vertical  plane  of  projection,  and  the  part  below  it  the  horizontal 
plane  of  projection. 

14.  Elementary  geometry  shows  that  the  plane,  as  PP'  p"2'>, 
PI.  I.,  Fig.  3,  of  the  projecting  lines,  Vp  and  PP',  (3,  4)  is  per- 
pendicular to  both  of  the  planes  of  projection,  and  to  the  ground 
line  MG.  Hence  it  intersects  these  j^lanes  in  lines,  as  j^lJ"  and 
P'  p",  both  of  which  arc  i)crpendicular  to  the  ground  line  at  the 
same  j^ointjy". 

15.  If,  now,  as  exiDlained  in  (13)  the  vertical  plane  MY,  PI.  I., 
Fig.  3,  be  revolved  about  MG,  to  coincide  with  the  horizontal 
plane,  the  point  p"  will  remain  in  the  axis  MG,  and  the  lines  23'j>" 
and  V'2}"  will  unite  to  form  one  line  2^p'>  perpendicular  to  MG. 

That  is:  Whenever  two  points  are  the  projections  of  one  point 
tJisjMce,  the  line  joining  them  iviU  he  perpendicular  to  the  ground 
line. 

§  Y. — Of  the  Conventional  Direction  of  the  Light ;  and  of  the 
Position  and  Use  of  Heavy  Lines. 

10.  Without  going  into  this  subject  fully,  as  in  Div.  III.,  it  is 
suflicicnt  to  say  here  that,  as  one  faces  the  vertical  plane  of  ])ro- 
jection,  tlic  light  is  assumed  to  come  from  behind,  and  over  the 
left  shoulder,  in  such  a  direction  that  eacli  2)rojectio7i  of  a  rag 
(but  not  the  ray  itself)  malces  an  ayigle  ofAo°  with  theground  line, 
as  shown  in  PI.  I.,  Fig.  G.  And  note  that  the  light  is  supposed 
to  turn  vuth  the  observer,  as  he  turns  to  face  any  other  vertical 
plane. 

17.  T\\Q  practical  effect  ot  the  preceding  assumption  in  refer- 
ence to  the  light,  is,  that  upon  a  body  of  the  form  and  position 
fihown  in  PI.  I.,  Fig.  5,  for  example,  the  top,  front,  and  left 
liand  surfaces — i.  e.  the  three  seen  m  the  Fig. — are  illuminated, 
wliile  the  other  three  faces  of  the  body  are  in  the  shade. 


FIRST   PRINCIPLES.  7 

18.  The  practical  rule  by  Avliicli  tlie  diroction  of  {\h\  liglit,and 
is  effect, are  indicated  in  the  projections,  is,  tliat  :ill  those  visible 

ed<jes  of  the  body  in  space,  which  divide  tlie  light  from  tlie  daik 
suriaces,  are  made  heavy  in  projection. 

19.  To  ilhistrate  :  Tlic  edges  BC  and  CD  of  the  body  in  sj^aco, 
Pi.  I.,  Fig.  5,  divide  liglit  from  dark  sui-faces,  and  are  seen  in 
looking  towards  the  vertical  plane,  and  hence  aro  made  heavy  in 
vertical  projection,  as  seen  at  h'c'  and  c'd'.  BK  and  KF  divide 
illuminated  from  dark  surfaces,  and  are  seen  in  looking  towards 
the  horizontal  plane,  and  are  therefore  made  heavy  in  the  hori- 
zontal ])rojection,  as  shown  at  bh  and  kf. 

20.  By  inspection,  it  will  be  seen  that  the  following  simple  rule 
in  reference  to  the  position  of  the  heavy  lines  on  the  drawings,  may 
be  deduced,  as  an  aid  to  the  memory.  In  all  ordinary  four-sided 
prismatic  bodies,  placed  with  their  edges  I'espectively  parallel  and 
perpendicular  to  the  planes  of  projection,  or  nearly  so,  the  right 
liand  lines,  and  those  Jiearest  the  ground  line,  of  both  ^projections, 
are  ynade  heavy. 

21.  Heavy  lines  are  of  considerable  use,  in  the  case  of  line  draw- 
ings particularly,  in  indicating  the  forms  of  bodies,  as  will  be  seen 
ill  future  examples.  In  shaded  drawings,  the  student  must  be 
careful  to  omit  the  heavy,  or  "shade  lines,"  which  habit,  in  mak- 
ing many  line  drawings,  might  lead  him  to  add.  On  flat  colored 
surfaces  tliey  should  be  added  last,  to  avoid  washing  them,  when 
coloring. 

§  VI. — Notation. 

22.  Under  the  head  of  Notation,  two  points  are  to  be  consi- 
dered,  the  manner  of  indicating  the  various  lines  of  the  diagram, 
and  the  lettering.  As  will  be  seen  by  examining  PI.  I.,  Figs.  1,  2 
— see  Ye,  eg,  &c. — and  5,  the  visible  lines  of  the  object  represented 
are  indicated  by  fidl  lines  ;  lines  of  construction  and  invisible  lines 
of  the  object,  so  far  as  they  are  shown,  are  made  in  dotted  lines 
The  intersections  of  auxiliary  planes  wiih  the  planes  of  projection 
called  traces,  are  represented  by  broken  and  dotted  lines,  as  at 
PQP',  PI.  I.,  Fig.  16. 

23.  Unaccented  letters  indicate  the  horizontal  projections  of 
points.  The  same,  witli  one  or  more  accents,  denote  their  vertical 
Itrojections.  The  sim])le  rule  of  thus  ahoays  lettering  the  sanu 
point  vrith  the  same  letter,  wherever  it  is  shown,  affords  a  key  to 
every  diagram,  as  will  be  shown  as  the  course  proceeds. 

The  projections  of  a  body  are  geometrically  equivalent  to  the 


FIRST    PRINCIPLES. 


body  itself,  since  they  show  its  form,  posifion  and  dimensions. 
Hence  objects  ure  considered  as  named  by  naming  their  projections. 
Thus, the  point ^7/;'  means  the  point  Avhose  projections  are/?  and/)'; 
tlie  line  ali — a'V  means  the  one  Avhose  projections  are«J  and  a'lj' . 
For  brevity,  the  horizontal  and  vertical  planes  of  projection  are 
designated,  respectively,  as  II  and  \ . 

•  24.  In  the  practical  a])plicationB  of  projections,  "horizontal  piO' 
jections"  are  usually  called  ^'■plans^''  and  ''vertical  projections," 
"  eUvationsr 

Before  entering  u])on  the  study  of  the  subsequent  constructions, 
the  terms  '•'■perpendicular''''  ^\\^ '•'•  verticaV  should  be  clearly  dis- 
tinguished. "  Perpendicular"  is  a  relative  terrn^  showing  that  any 
line  or  surface,  to  which  it  is  applied,  is  at  right  angles  to  sotno 
other  line  or  surface.  "  Vertical"  is  an  absolute  term^  at  any  one 
place,  and  applies  to  any  line  or  surface  at  right  angles  to  a  level, 
as  a  water  surface.  A  vertical  line,  L,  is  perpendicular  to  all  hori- 
zontal lines  which  intersect  it,  but  if  the  entire  system  of  lines  thus 
related  were  inclined,  so  that  all  should  be  oblique,  L  would  still 
be  perpendicular  to  all  the  rest,  though  no  longer  vertical. 

§  VII.— 0/  the  Use  of  the  Method  of  Projections. 

25.  Under  this  head  it  is  to  be  noticed,  that  all  drawings  are 
made  to  serve  one  or  the  other  of  two  purposes,  i.e.  they  are  made 
for  xcse  in  aiiling  workmen  in  the  construction  of  works  ;  or  in 
rendering  intelligible,  by  means  of  drawings,  the  real  form  and  size 
of  some  existing  structure ;  or  else,  they  are  made  for  ornament^ 
or  to  embellish  our  houses  and  gratify  our  tastes,  and  to  show  the 
a2>parent  forms  and  relative  sizes  of  objects. 

2G.  Drawings  of  the  former  kind  are  often  called,  on  account  of 
the  uses  to  which  they  are  applied,  '•'■mechanical''^  or  '■'•  xoork'mg '''' 
drawings.  Those  of  the  latter  kind  are  commonly  called  pictures  ; 
and  here  it  is  to  be  noticed  that  if  "working"  drawings  are  to 
Bhow  the  tnie^  and  not  the  upjjxtreiU^  proportions  of  all  parts  of  an 
object,  they  must,  all  and  always,  conform  to  this  one  rule,  viz. 
All  those  lines  wldch  are  equal  and  similarly  situated  on  the  oljoi't, 
must  be  equal  and  similarly  situated  on  the  drawing. 

But,  as  is  now  abundantly  evident,  drawings  made  according  to 
the  method  of  projections,  do  conform  to  this  rule;  hence  their  use, 
a?  aboN  e  described. 


CHAPTER  n. 

PROJECTIONS  OF  LINES:  PROBLEMS  IN"  RIGHT  PROJECTION;  AND 
INCLUDING  PROJECTIONS  SHOWING  TWO  SIDES  OF  A  SOLID 
RIGHT    ANGLE. 

27.  The  style  o?  execution  of  the  following  problems  is  so  simple, 
and  so  nearly  alike  for  all  of  them,  that  it  need  not  be  described 
for  each  problem  sepai'ately,  but  will  be  noticed  from  time  to  time. 
In  the  solution  of  problems,  Imes  are  considered  as  wdimiied,  and 
may  be  produced  mdefij^itely  in  either  direction. 

§  I. — Projections  of  Straight  Lines. 

28.  Prob.  1.  To  construct  the  jyojections  of  a  vertical  straight 
line,  1^  inches  long,  xohose  lowest  point  is  \  an  inch  from  the 
horizontal  plane,  and  all  of  whose  points  are  ^  of  an  inch  from 
the  vertical  plane. 

Jiemarks.  a.  The  remaining  figures  of  PI.  I.  are  drawn  just 
half  the  size  indicated  by  the  dimensions  given  in  the  text.  It 
may  be  well  for  the  student  to  make  them  of  full  size. 

b.  Let  MG  be  understood  to  be  the  ground  line  for  all  of  the 
above  problems,  without  further  mention  of  it. 

1st.  Draw,  very  lightly,  an  indefinite  line  perpendicular  to  the 
ground  line,  PI.  I.,  Fig.  7. 

2d.  Upon  it  mark  a  point,  a',  two  inches  above  the  ground  line, 
and  another  pohit,  6',  half  an  inch  above  the  ground  line. 

2d.  Upon  the  same  line,  mark  the  point  a,b,  three-fourths  of  an 
lich  below  the  ground  line.  Then  a' i' will  be  the  vertical,  and 
ab  the  horizontal  projection  of  the  required  line.  (8  a) 

29.  Prob.  2.  To  construct  the  projections  of  a  horizontal  line, 
H  inches  long,  Ij  inches  above  the  horizontal  plane,  perpendicular 
to  the  vertical  plane,  andxoith  its  furtherm,ost  point — from  the  eye 
— \  of  an  inch  from  that  j)lane.  PL  I.,  Fig.  8,  in  connection  with 
the  full  description  of  the  preceding  problem,  will  afford  a  sufficient 
explanation  of  this  one. 

Remark.     It  often  happens  that  a  diagram  is  made  more  intel 


li)  TKOJECTIOXS    OF    LINES 

ligiUe  by  lettering  it  as  at  aZ»,  PI.  I.,  Fig.  *",  and  at  c'd\  PI.  I.  Fig 
8,  for  thus  the  notation  shows  unmistakably,  that  ah  or  c'd'  are  not 
the  projections  of  points  but  of  lines. 

30.  PnoBLEMS  3  to  8,  inclusive,  need  now  only  to  be  enunciated 
witli  references  to  their  constructions,  in  PI.  I. 

Fig.  9  shows  the  projections  of  a  line,  2\  inches  long,  parallel  tc 
the  ground  line,  H  inches  from  the  horizontal  plane,  and  1  inch 
from  the  Acrtical  plane. 

Fig.  10  is  the  representation  of  a  line,  2  inches  long ;  parallel  to 
the  horizontal  piano,  and  1  inch  above  it ;  and  making  an  angle  of 
30°  with  the  vertical  plane. 

Fig.  11  represents  a  line,  2^  inches  long,  i)arallel  to  the  vertical 
plane,  and  1|  inches  from  it,  and  making  an  angle  of  60°  with  the 
horizontal  plane. 

Fig.  12  gives  the  projections  of  a  line,  1^  inches  long,  lying  in 
the  horizontal  plane,  parallel  to  the  ground  line,  and  1^  inches 
from  it.  The  jirojcction  a'h'  shows  the  lino  to  be  in  II  (G,  ^tli). 

Fig.  13  shows  the  projections  of  a  line,  1^  inches  long,  lying  in 
the  vertical  plane,  parallel  to  the  ground  line,  and  1  inch  from  it. 

Fig.  14  indicates  a  line,  2^  inches  long,  lying  in  the  vertical 
plane,  and  making  an  angle  of  60°  with  the  horizontal  plane. 

31.  Projections  of  the  revolution  of  a  2^oint  about  an  axi- 
When  a  point  revolves  about  an  axis,  it  describes  a  circle,  or  arc, 
whose  plane  is  perpendicular  to  the  axis.  Tiius  a  point,  revolving 
al)out  an  axis  which  is  pe7'pendicular  to  the  vertical  plane,  describes 
an  arc,  parallel  to  that  plane.  The  vertical  projection  of  such  an 
arc  is  an  equal  arc.  Its  horizontal  projection  (6)  is  a  straight  line 
parallel  to  the  ground  line. 

Tims,  Pi.  I.,  Fig.  loa,  Ca  represents  a  perpendicular  to  the  ver 
tical  plane,  V.  Tlie  point,  A,  by  revolving  a  certain  distance  about 
this  axis,  describes  the  arcAB;  whose  vertical  projection  is  the 
iq>ial  arc,  a'b';  and  whose  horizontal  projection  is  a5,  a  straight 
line  parallel  to  the  ground  line. 

Likewise,  briefly,  in  Figs.  \5b  and  15c,  XYjs  a  vertical  axis. 
The  point  A,  revolving  abcmt  it,  describes  a  horizontal  arc,  AB; 
wliose  horizontal  projection,  ai,  is  an  equal  arc  ;  and  whose  verti- 
cal projection, a'i'  is  a  straight  line  jjarallel  to  the  ground  line. 

32.  PiiOH.  9.  7b  construct  tliC  jyrojections  of  a  line  lohich  is  i?i  a 
plane  perpendicular  to  both  pAanes  of  projection^  the  line  being 
oblique  to  both  planes  of  j/rojection.     Plate  I.,  Fig.  15,  represents  a 


PROJECTION  OF   LINES.  11 

model  of  this  problem,  AB  represents  the  line  in  space;  ab  iln 
horizontal  projection;  a'b'  its  projection  on  the  vertical  plane  MP'; 
and  A'B'  its  projection  on  an  auxiliary  vortical  plane  PQP';  which 
is  parallel  to  AB,  and  perpendicular  to  the  ground  line.  Ilenoe 
A'B'=AB. 

Now  in  making  these  three  planes  of  projection  coincide  with 
the  paper,  taken  as  the  horizontal  plane  of  projection,  the  plane 
PQP'  is  revolved  about  P'Q  as  an  axis,  till  it  coincides  with  the 
primitive  vertical  plane,  MP',  produced,  as  at  P'QV",  and  then  the 
united  vertical  planes,  MPV",  are  revolved  backward  about  MH' 
as  an  axis  into  the  horizontal  plane.  In  the  first  revolution.  A' 
describes,  according  to  the  last  article,  the  horizontal  arc,  A'a", 
(31)  about  m  as  a  centre,  and  whose  projections  are  «""«'",  liaving 
its  centre  at  Q,  and  wia".  Also  B'  describes  the  horizontal  arc, 
B'J",  about  71  as  a  centre,  and  whose  projections  are  h""h"\  whose 
centre  is  Q,  and  nh" .  Thus  wc  see  that  two  or  more  differe^it  vev' 
tical 2^roJections,  as  a'  and  a",  of  the  same  j^oint,  are  in  the  same 
jmrallel,  a'a",  to  the  ground  line;  that  is,  the^/  are  at  the  same 
lieight  above  that  line.  Hence  a"  is  at  the  intersection  of  a'a", 
parallel  to  the  ground  line,  MIP,  with  «'''«'■',  perpendicular  to  MH'. 

33.  a.  Notice  further  that  a""h""  is  the  horizontal  projection  of 
A'B',  and  that  it  coincides  with  the  projection  of  ab  upon  PQP' 
Likewise,  that  mn  is  the  vertical  projection  of  A'B',  and  that  it  co 
incides  with  the  projection  oi  a'b'  upon  PQP'. 

b.  Note  that  B^',  for  example  (6),  is  equal  to  bt,  and  that  bt=b"n 
the  distance  of  the  auxiliary  vertical  projection,  b",  of  B,  from  the 
trace,  or  axis,  P'Q,  of  the  auxiliary  plane. 

c.  Note  that  a"b"  shows  the  true  length  and  direction  o?  A\^ ; 
that  is,  the  angles  made  by  a"b"  with  ll'Q  and  P'Q,  respectively, 
are  equal  to  those  made  by  AB  with  the  planes  of  projection. 

34.  To  construct  PL  I.,  Fig.  15,  in  2^yojection.  See  Pi.  L,  Fig. 
IG,  where,  to  make  the  comparison  easier,  like  points  have  the 
same  letters  as  in  Fig.  15.  Supposing  the  length  and  direction  of 
the  line  given,  we  begin  with  a"b",  which  suppose  to  be  2"  long, 
and  to  make  an  angle  of  60''  with  the  horizontal  plane.  Suppose 
the  line  in  space  to  be  \\  inches  to  the  left  of  the  auxiliary  vertical 
plane  P'QP  then  a'  b',  its  vertical  projection,  will  be  perpendicular 
to  the  ground  line,  between  the  i>arallels  a"a'  and  b"b'  (32),  and  U 
inches  from  P'Q.  The  horizontal  projection,  ah,  will  be  in  a'b' 
produced;  b"n — b"'b""  are  the  two  projections  of  the  arc  in  which 
ihe  point  b"  revolves  back  to  its  position,  n — b"",  in  the  plane 
P'QP,  and  b""b — nb'  is  the  line  in  which  nb""  is  projected  back 


12  PBOJECnOK   OF  LTSTES. 

t 

to  it5  primitive  position  h'b.  Therefore,  h  is  at  the  intersection 
of  i""6  with  afb'  proiiuced.  a  is  similarly  found,  giving  ab  as  the 
liorizontal  projection  of  the  given  line. 

An.  32  shows  suflSciently  how  to/V«t?  the  length  a^'b"  \£ ab — a^b' 
were  given. 

Example.     Construct  the  figure  when  a',b  is  the  highest  point. 

35.  Exf.cution.  The  foregoing  problems  are  to  be  inked  with 
very  black  ink;  the  projections  of  given  lines,  and  the  ground  line, 
m.  heavy  full  lines;  and  the  lines  of  construction  vafine  dotted  lines 
as  shown  in  the  figures.  Lettering  is  not  necessary,  except  for 
purposes  of  reference,  as  in  a  text  book,  though  it  affords  occasion 
for  practice  in  making  small  letters. 

On  the  other  hand,  lettering,  if  poorly  executed,  disfigures  a 
diagram  so  much  that  it  should  be  made  only  after  some  pre  vie  ua 
pract.oe,  and  then  carefiilly ;  making  the  letters  small,  fine,  and 
regular. 

§  IL — Right  Projections  of  Solid*, 

Remark.  The  tenn  "  right  projection  "  becomes  significant  only 
when  it  refers  to  bodies  which  are,  to  a  considerable  extent, 
bounded  by  straiglit  lines  at  right  angles  to  each  other.  Such 
bodies  are  said  to  be  drawn  in  right  projection  when  their  most 
important  lines,  and  faces,  are  parallel  or  perpendicular  to  one  or 
the  other  of  the  planes  of  projection. 

36.  Peob.  10. —  To  construct  the  projections  of  a  vertical  right 
prism,  having  a  square  base;  standing  upon  the  horizontal  plane, 
and  icith  one  of  its  faces  paralltl  to  the  vertical  plane.  PI.  U., 
Fig.  17- 

Let  the  prism  be  1  inch  square,  H  inches  high,  and  ^  of  an  inch 
from  the  vertical  plane. 

\st.  Tiie  square  ABEF,  \  of  an  inch  from  the  ground  line,  is  the 
plan  of  the  prism,  and  stiijtly  represents  its  upper  base. 

2d.  A'B'C'D',  1^  inches  high,  is  the  elevation  of  the  prism,  and 
strictly  represents  its  front  face. 

Ill  this,  and  in  all  similar  problems,  it  is  useful  to  distinguish  the 
positions  of  the  p'jints,  lines,  and  faces,  in  words ;  as  ripper  aud 
lotcer /  front  and  back  /  right  and  left /  just  as  is  done  in  speaking 
of  the  bodies  which  (23)  the  projections  represent.     Thus, 

Ist.   AA'  is  the  fn)nt,  upper,  left  hand  comer  of  the  prism. 

2d.  EF — A'C  i.s  the  back  top  edge  ;  BF — D'  is  the  lower  riglit 
hand  edge  ;  each  corner  of  the  plan  is  the  horizontal  projection  of 
a  vertical  edge  ;  etc. 


PIi.1. 


c 


PROJECTIONS  OP   SOLIDS.  13 

Sd.  AE — A'C  is  the  left  hand/ace;  etc. 

37.  Prob.  11. — To  construct  the  plan  and  two  elevations  of  a 
vrism  having  the  proportions  of  a  brick,  and  placed  with  its  length 
(>a7'allel  to  the  ground  line.     Plate  II.,  Fig.  18. 

Ist.  abed  is  the  plan,  f  of  an  inch  broad,  twice  that  distance  in 
length,  and  f  of  an  inch  from  the  ground  line,  showing  that  the  prism 
in  space  is  at  tlie  same  distance  from  the  vertical  plane  of  projection. 

2nd.  a'b'ef  is  the  elevation,  f  of  an  inch  thick,  and  as  long 
as  the  plan  ;  and  ^  of  an  inch  above  the  ground  line,  showing  that 
the  prism  in  space  is  at  this  height  above  the  horizontal  plane. 

^rd.  If  a  plane,  P'QP,  be  placed  jDcrpendicular  to  both  of  the 
principal  planes  of  projection,  and  touching  the  right  hand  end 
of  the  prism,  it  is  evident  that  the  projection  of  the  prism  upon 
such  a  plane  will  be  a  rectangle,  equal,  in  length,  to  the  width,  bd^ 
of  the  plan,  and,  in  height,  to  the  height,  bf  of  the  side  elevation. 
This  new  projection  will  also,  evidently,  be  at  a  distance  from  the 
primitive  vertical  plane,  i.e.  from  P'Q,  equal  to  JQ,  and  at  a  dis- 
tance from  the  horizontal  plane  equal  to  Q/".  When,  therefore, 
the  auxiliary  plane,  P'QP,  is  revolved  about  P'Q  into  the  primitive 
vertical  i^lane  of  j^rojection,  the  new  projection  will  appear  at 
a"e"c"g". 

ith.  dc'"  is  the  horizontal,  and  b'o"  the  vertical  projection  of 
the  arc  in  which  the  point  db'  revolves  into  the  primitive  vertical 
plane.  Ja'",  b'a%  are  the  two  projections  of  the  horizontal  arc  in 
which  the  corner  bb'  of  the  prism  revolves. 

Example. — Let  the  auxiliary  plane  PQP'  be  revolved  about  PQ 
into  the  horizontal  plane.  a"c"  will  then  appear  to  the  right  of 
PQ  and  at  a  distance  from  it  equal  to  Qb'. 

38.  Prob.  12. — To  const7'uct  the  two  2y'>'ojections  of  a  cylinder 
which  stands  upon  the  horizontal  jolane.     PI.  IL,  Fig.  19. 

The  circle  AaB5  is  evidently  the  plan  of  such  a  cylinder,  and 
the  rectangle  A'B'C'D'its  elevation.  Observe,  here,  that  while  the 
elevation,  alone,  is  the  same  as  that  of  a  prism  of  the  same  height. 
Fig.  17,  tlie  plan  shows  the  body  represented,  to  be  a  cylinder. 

Any  point  as  a  in  the  plan,  is  the  horizontal  projection  of  a  ver 
tical  line  lying  on  the  convex  surface,  and  called  an  element.  A — 
A'C,  and  B — B'D',  which  limit  tlie  visible  part  of  the  convex  sur- 
face, are  called  the  extreme  elements. 

39.  As  regards  execution,  the  right  hand  line  B'D'  of  a  cylindef 


14  PROJECTIONS   OK   SOLIDS. 

or  cone  may  l)e  made  less  Iieavy  than  tlie  line  B'D',  Fig,  IV  ;  and 
in  the  plan,  the  semicircle,  aVtb,  convex  towards  the  ground  line, 
and  limited  by  a  diameter  ab,  which  makes  an  angle  of  45°  with 
the  ground  line,  is  made  heavy,  but  gradually  tapered,  into  a  fine 
line  in  the  vicinity  of  the  points  a  and  b, 

40.  Prob.  13. — To  construct  the  projections  of  a  cylinder  whose 
axis  is  2)lciced  parallel  to  the  f/round  line.     PI.  11.,  Fig.  20. 

Let  the  cylinder  be  1^  inches  long,  f  of  an  inch  in  diameter,  its 
axis  f  of  an  inch  from  the  horizontal  plane,  and  ^  an  inch  from  the 
vertical  plane.  The  principal  projections  will,  of  course,  be  two 
equal  rectan<:^les,  geh/  and  a'h'c'd\  since  all  the  diameters  of  the 
cylinder  are  equal.  The  centre  lines,  cfh'  and  a5,  are  made  at  the 
same  distances  from  the  ground  line,  that  the  axis  of  the  cylinder 
is  from  the  planes  of  projection  (G). 

The  end  elevation,  knowing  its  radius,  which  is  equal  to  half  of 
the  diameter  ge^  or  aV,  of  the  cylinder,  may  be  made  by  revolving 
the  jirojection  of  its  centre  a/j\  only,  upon  PQP',  around  P'Q  as 
an  axis. 

41.  Standing  with  a  horizontal  cylinder  before  one,  with  its  axis 
lying  from  right  to  left,  and  parallel  to  the  ground  line,  one  of  its 
elements  is  its  highest  one,  that  is  the  highest  above  the  ground,  or 
the  horizontal  plane  ;  another  is  the  loicest ,'  another,  i\\Q  foremost^ 
that  is  the  one  nearest  to  one,  and  another  the  hindmost,  or  the  one 
furthest  from  one.  Ti-ansfering  the  same  terms  to  the  jyrojections 
of  the  same  elements,  by  (23)  we  have  ab — a'  h' — h"  [the  three  pro- 
jections of  j  the  higliest  element;  ab — c'd' — c?'',  the  loxoest  element; 
(jh — g'h' — /i",  the  foremost  element;  and  ef — g'Ji' — -/*",  the  hiiid- 
most  element. 

In  inking,  the  end  elevation,  b"f"d",  is  made  heavy  at  nf"d"p, 
and  tapered  into  a  fine  line  in  the  vicinity  of  n  and  ^9/  because,  by 
(16)  when  the  observer  turns  to  face  the  plane  PQP',  looking  at  it 
ill  the  direction  hg  (12),  the  light  turns  with  him. 

42.  We  have  now  three  ways  of  distingui-;iiing  the  projections 
of  a  horizontal  cylinder  from  those  of  a  square  prisin  of  equal 
dimensions.  First,  Ijy  medium  instead  of  fully  heavy  lines  on 
ff  and  c^d'.  Seco)id,hy  the  lettering  of  the  principal  elements,  a? 
just  explained.  Third,  and  most  clearly,  by  the  circular  end  ele 
vation. 

§  in — Projections  showing  two  sides  of  a  Solid  Right  Angle. 
43    A  solid  right  angle  is  an  angle  such  as  that  at  any  corner  of 


7& 


2D. 


pr.if. 


■b" 


A'      22.        B"  A'  2S.     B"       B"'0"-  26!     D" 


E^' 


F"  P'"  G 


d 


25. 


!»•  P'         f" 


:     27. 


PROJECTIONS   OF   SOLIDS.  If, 

A  cube,  or  a  square  prism,  and  is  therefore  bounded  by  three  plane 
riglit  angles.  When  two  faces  of  such  a  body  are  seen  at  once 
they  will  be  seen  obliquely,  and  neither  will  appear  in  its  true  size. 
Hence  only  o?ie  of  the  projections  of  the  object  will  show  iioo  o( 
its  dimensions  in  their  real  size.  Hence,  we  must  always  malie 
first,  that  projection^  whichever  it  be,  which  shotcs  two  dimensious 
in  their  real  size. 

44.  Peob.  14. — To  construct  the  plan  and  ttco  elevations  of  a 
Vtyrtical  jyrism,  xoith  a,  square  base  ;  resting  on  the  horizontal  2^1  ane, 
and  having  its  vertical  faces  inclined  to  the  vertical  plane  of  pro- 
jection.   PL  n.,  Figs.  21 — 22. 

\st.  ABCG  is  the  plan,  wdiich  must  be  made  first  (43)  and  with 
its  sides  placed  at  any  convenient  angle  with  the  ground  line. 

Id.  A'B'D'E'  is  the  vertical  projection  of  that  vertical  face 
whose  horizontal  projection  is  AB. 

3f?.  B'C'E'F'  is  the  vertical  projection  of  that  face  whoso  hori- 
zontal projection  is  BC.  This  completes  the  vertical  projection 
of  the  visible  parts  of  the  prism,  when  we  look  at  the  prism  in  the 
direction  of  the  lines  CF',  &c. 

4?/t.  Let  gh  be  the  horizontal  trace  of  an  auxiliary  vertical  plane 
of  projection,  which  is  perpendicular  to  both  of  the  principal  j^lancs 
of  ])rojcction.  In  looking  perpendicularly  towards  this  jilane,  i.e. 
/n  the  directions  G^,  &c.,  AG  and  AB  are  evidently  the  hoiizontal 
projections  of  those  vertical  faces  that  would  then  be  visible;  and 
the  projecting  lines,  Gy,  Aa,  and  B^  determine  the  widths  ga  and 
ah  of  those  faces  as  seen  in  the  new  elevation.  Now  the  auxiliary 
plane  gh  is  not  necessarily  revolved  about  its  vertical  trace  (not 
shown),  but  may  just  as  well  be  taken  up  and  transferred  to  any 
position  where  it  will  coincide  with  the  primitive  vertical  plane : 
only  its  ground  line  gh  must  be  made  to  coincide  with  the  principal 
ground  line,  as  at  WYJ' .  Hence,  making  H"D"  and  D"£"  respec- 
tively equal  to  ga  and  ah,  and  by  drawing  H"G",  &c.,  the  new 
elevation  will  be  completed. 

45.  The  two  elevations — PI.  H.,  Figs.  21,  22 — appear  exactly 
alike,  but  the  faces  seen  in  Fig.  22  are  not  the  same  as  the  equa. 
ones  of  Fig,  21. 

The  different  projections  of  the  same  face  maybe  distinguished 
by  mnrks.  Thus  the  surfaces  marked  1^  are  the  two  elevations  of 
Ihe  same  face  of  the  prism;  the  one  maiked  ^  is  visible  only  on  the 
first  elevation,  and  the  one  marked  x  is  visible  only  on  the  second 
felevation — Fi^.  22. 


16  TBOJECTIONS    OF   SOLIDS. 

46.  PI.  II.,  Fig.  23,  represents  a  small  quadrangular  prism  in  twc 
elevations,  the  axis  being  horizontal  in  space,  so  that  the  left  hand 
elevation  shows  the  base  of  the  ])rism.  In  the  practical  applica- 
tions of  this  construction,  the  centre,  s,  of  the  square  projection  is 
generally  on  a  given  line,  not  parallel  to  the  sides  of  the  square. 
Hence  tliis  construction  afibrds  occasion  for  an  application  of  the 
problem  :  To  draw  a  square  of  given  size,  with  its  centre  on  a  given 
line,  and  its  sides  7iot  jxirallel  to  that  line.  The  following  solution 
should  be  carefully  remembered,  it  being  of  frequent  application. 
Through  the  given  centre,  s,  draw  a  line,  L,  in  any  direction,  and 
another,  L',  also  through  s,  at  right  angles  to  L.  On  each  of  these 
lines,  lay  off  each  way  from  5,  half  the  length  of  a  side  of  the 
square.  Through  the  points  thus  formed,  draw  lines  parallel  to 
the  lines  L  and  L'  and  they  will  Ibrm  the  required  square  whose 
centre  is  s. 

47.  Proi5.  15. — Ih  construct  the  plan  and  several  elevations  of  a 
vertical  hexagonal  prism,  lohich  rests  upon  tlie  horizontal  plane  oj 
projection.     PI.  II.,  Figs.  24,  25,  26. 

Tlie  distinction  between  bodies  as  seen  perpendicularly,  or  ob- 
liquely, becomes  obscure  as  we  pass  from  the  consideration  of 
bodies  whose  surfaces  are  at  right  angles  to  each  other.  Figs.  24 
and  25  show  a  hexagonal  prism  as  much  in  right  prrvjection  as  such 
a  body  c:in  be  thus  shown,  but,  as  in  both  cases  a  majority  of  ita 
surfaces  aie,  considered  separately,  seen  obliquely,  its  construction 
is  given  here. 

In  Fig.  24  the  hexagonal  prism  is,  as  shown  by  the  ]ilan,  placed 
so  that  two  of  its  vertical  faces  are  parallel  to  the  vertical  plane 
of  ])rojection.  Observe  that  where  the  hexagon  is  thus  placed, 
three  of  its  faces  will  be  visible,  one  of  them  in  its  real  size,  viz., 
r>C,  B'C'F'G',  and  that  the  extreme  width,  E'tP,  of  the  eleva- 
tion, equals  the  diameter,  AD,  of  the  circumscribing  circle  of  the 
plan.  This  is  therefore  tlie  loidest  possible  elevation  of  this  prism. 
Xolicc,  also,  that  as  BC  equals  half  of  AD,  while  AB  and  CD  are 
equal,  and  equally  inclined  to  the  vertical  plane,  the  elevations,  A'F' 
and  G'D',  of  these  latter  faces,  %oill  he  equal,  and  each  half  as  wide 
as  the  middle  face.  This  Ihct  enables  us  to  construct  the  elevation 
of  a  hexagonal  prism  situated  as  here  described,  without  construct- 
ing the  plan,  provided  we  know  the  width  and  height  of  one  face 
of  the  prism.  This  last  construction  should  be  remembered,  i1 
being  of  fi-cquent  and  convenient  application  in  the  drawing  of 
Quts,  bolt-heads,  tfec,  in  machine  drawing. 


PBOJECTIONS    OF    SOLIDS.  11 

48.  PI.  n.,  Fig.  25  shows  the  elevation  of  the  same  prism  on  a 
plane  which  originally  was  placed  at  ib^  and  perpendicular  to  the 
horizontal  plane  ;  whence  it  nppears,  that  if  a  certain  elevation  of  a 
hexagonal  prism  shows  three  of  its  faces,  and  one  of  them  in  its 
full  size,  anotlier  elevation,  at  right  angles  to  this  one,  will  show 
but  two  faces,  neither  of  them  in  its  full  size  ;  the  extreme  width, 
I"B",  of  the  second  elevation  being  equal  to  the  diameter  of  tho 
uiscribed  circle  of  the  plan.  This  is  therefore  the  narrowest  pos- 
sible elevation  of  this  prism. 

49.  PI.  II.,  Fig.  26  shoios  the  elevation  of  the  same  prism  as  it 
a}ypears  when  projected  upon  a  vertical  plane  standing  on  jb'%  and 
then  transferred  to  the  principal  vertical  plane,  at  Fig.  26.  In  this 
elevation,  none  of  the  faces  of  the  prism  are  seen  in  their  true  size. 
The  auxiliary  vertical  plane,  owjb",  could  have  been  revolved  about 
that  trace,  directly  back  into  the  horizontal  plane,  causing  the 
corresponding  elevation  to  Appear  in  the  lines  Df?,  &c.,  produced 
to  the  left  of  jV  as  a  ground  line.  Elevations  on  auxiliary  vertical 
planes  can  always  be  made  thus,  but  it  seems  more  natural  to  see 
them  side  by  side  above  the  principal  ground  line,  by  transferring 
the  auxiliary  planes  as  heretofore  described. 

50.  Fig.  27  represents  two  elevations  of  a  hexagonal  prism, 
placed  so  as  to  show  the  base  in  one  elevation,  and  three  of  its 
faces,  unequally,  in  the  other.  The  centre  of  the  elevation  which 
shows  the  base,  may  be  made  in  a  given  line  perpendicular  to  o'fj\ 
by  placing  the  centre  of  the  circumscribing  circle  used  in  con- 
structing the  hexagon,  upon  such  a  Ihie.  Having  constructed  this 
elevation,  project  its  points,  a,6,  &c.,  across  to  the  other  vertical 
plane,  P',  which  is  in  space  perpendicular  to  the  plane,  P,  at  the 
line,  o'g'.  By  representing  the  elevation  on  P'  as  touching  o'g\  we 
indicate  that  the  prism  touches  the  plane,  P,  just  as  the  elevation 
in  Fig.  24,  indicates  that  the  prism  there  shown  rests  upon  the 
horizontal  plane. 

51.  Peob.  16. — To  construct  the  plan  and  two  elevations  of  a 
pile  of  blocks  of  equal  widths,  but  of  different  lengths,  so  placed 
as  to  form  a  symmetrical  body  of  uniform  width.  PI.  HI., 
Figs.  28,  29. 

Here  for  example  afg  is  the  plan  of  the  lowest  step;  kbe  is  that 
of  the  middle  step,  and  cdh  that  of  the  upper  step  (43). 

The  auxiliary  vertical  plane  of  projection,  perpendicular  to  the 
horizontal  plane  at  hf"f"\  is  made  to  coincide  with  the  principal 
vertical  plane  by  direct  revolution.     The  point  a'"a"'\  the  projec 


18  PROJECTIONS   OF   SOLIDS. 

tion  of  an'  on  the  auxiliary  vertiral  pl;ine,  revolrcs  in  a  liorlzonta) 
arc,  of  wliich  a"'a"  is  ihe  horizontal,  aiul  a""a*'  the  vertical  pro- 
jection (31),  giving  a",  a  point  of  the  second  elevation.  Other 
points  of  this  elevation  are  found  in  tlie  same  way.  This  figure 
differs  from  Figs.  18  and  20,  of  Plate  II.,  only  in  presenting  more 
points  to  be  constructed.  If  the  student  finds  any  difficulty  witli 
this  example,  let  him  refer  to  those  just  mentioned,  and  to  first 
principles. 

Example. — Construct  an  elevation  on  a  plane  parallel  to  af. 

52.  Peob.  17. —  To  constnict  the  vertical  projection  of  a  verticai 
circle,  seen  ohliquehj.    Pi.  III.,  Fig.  30. 

Let  BF  be  the  given  projection  of  the  circle.  It  is  required  to 
find  its  vertical  projection,  A'B'D'F'.  For  this  purpose,  the  circle 
must  be  first  brought  into  a  position  parallel  to  a  plane  of  projec- 
tion, since  we  can  then  make  both  of  its  projections,  and  hence  can 
then  take  both  projections  of  any  point  upon  it.  Let  the  circle  be 
made  parallel  to  the  vertical  plane.  To  do  this,  it  only  need  be 
revolved  about  any  vertical  axis.  In  the  figure,  the  axis  is  the 
vertical  tangent,  Y—f'V.  After  this  I'evolution,  the  projections 
of  the  circle  are  bV — i'c'FV/.  Now  taking  any  j)oint  on  this  circle, 
as  aa\  it  returns  about  the  axis  F — -f'Y'  in  the  horizontal  arc 
ff  A — a' A!  (31),  giving  A'  by  projecting  A  upon  a' h! .  Likewise  bb' 
returns  in  the  arc  bVt — J'B  to  BB';  and  cc',  which  is  vertically  under 
aa\  returns  in  the  arc  cC — c'C  to  CC.  Thus  all  the  points  A',B', 
C,  etc.,  being  found  and  joined,  we  have  A'B'C'II',  the  required 
oblique  elevation  of  the  vertical  circle  FB. 

Examples. — \st.  Let  the  circle  be  revolved  about  its  vertical 
diameter  IID,  or  any  vertical  axis  between  F  and  B. 

Id.  About  any  vertical  axis ;  in  the  plane  BF  produced ;  or  only 
parallel  to  it. 

Zd.  Let  the  ciicle  be  made  perpendicular  to  the  vertical  plane, 
and  oblique  to  the  horizontal  plane. 

63.  Prob.  18. — To  construct  the  projectiom  of  a  cylinder  whose 
convex  surface  rests  on  the  horizontal  plane,  and  whose  axis  is  in' 
dined  to  the  vertical  plane.     PI.  III.,  Fig.  31. 

As  may  be  learned  ivvm  Fig.  19,  PI.  II.,  the  projection  of  a  right 
cylinder  upon  any  plane  to  which  its  axis  is  parallel,  will  be  a 
rectangle.  Therefore  let  CSTV,  PI.  III.,  Fig.  31,  be  the  plan  of 
the  cylinder.  Since  it  nsts  uj)on  the  lioiizontal  ])lane,  q'u\  in  the 
ground  line,  is  the  vertical  j)rojection  of  its  line  of  contact  with  that 


PROJECTIONS   OP   SOI.ros.  Ij 

plane,  nnd^'A'is  the  vertical  projection  of  joA,  the  highest  element 
of  the  cylinder,  as  it  is  at  a  height  above  the  ground  line,  equal  to 
the  diameter,  TV,  of  the  cylinder.  The  vertical  projection  of  either 
base  may  be  found  by  the  last  problem.  In  the  figure,  the  left 
hafid  base  is  thus  found,  and  the  construction,  being  fully  given, 
needs  no  further  explanation. 

54.  The  vertical  projection  of  the  right  hand  base  TV  is  found 
somewhat  differently.  It  is  revolved  about  its  horizontal  diameter, 
TV — T'V,  till  parallel  to  the  horizontal  plane.  It  will  then  appear 
as  a  circle,  and  a  line,  as  n"n^  will  show  the  true  height  oin  above 
the  diameter  TV.  So,  also,  o"o  will  show  the  true  distance  of  o 
below  TV.  Therefore  the  vertical  piojections  of  the  points  n  and  o, 
will  be  in  the  line  n — n\  perpendicular  to  the  ground  line,  and  at 
distances  above  and  below  T'V,  the  vertical  projection  of  TV, 
equal,  respectively,  to  nn"  and  oo".  Having,  in  the  same  manner, 
found  t'  and  ;;',  the  vertical  projections  of  two  points  whose  com- 
mon horizontal  projection  t — r  is  assumed,  as  was  n — o,  the  vertical 
projection  of  the  base  TV  can  be  drawn  by  the  help  of  the  irregu- 
lar curved  ruler. 

55.  In  the  execution  of  this  figure,  SV  is  made  slightly  heavy,  and 
TV  fully  heavy,  and  the  portion,  n'T't\  of  the  elevation  of  the  right 
hand  base,  and  the  small  portion, DV,  of  the  left  hand  base,  are  made 
heavy.  Suffice  it  to  say :  First.  That  a  part  of  the  convex  surface 
is  in  the  light,  while  the  right  hand  base  is  in  the  dark.  *  Second. 
niT't'  divides  the  illuminated  half  of  the  convex  surface,  from  the 
base  at  the  right,  which  is  in  the  dark  ;  and  D'u'  divides  the  illu- 
minated left  hand  base  from  the  visible  portion  of  the  darkened 
htilf  of  the  convex  surfoce  (18-20). 

Example.  Let  the  axis  of  the  cylinder  be  parallel  to  the  vertical 
plane,  only. 

66.  Prob.  19.     To  construct  the  two  projections  of  a  right  cone 
with  a  circular  base  in  the  horizontal  plane  ;  and  to  construe 
either  p>rojection  of  a  line,  drawn  from  the  vertex  to  the  circum- 
ference of  the  base,  having  the  other  projection  of  the  same  lint 
given.    JPl.  III.,  Fig.  32. 

Remark.  When  the  axis  of  a  cone  is  vertical,  perpendicular  to 
the  vertical  plane,  or  parallel  to  the  ground  line,  the  cone  is  sho^vD 
in  right  projection  as  much  as  such  a  body  can  be,  but  as  all  the 
straight  lines  upon  its  surface  are  then  inclined  to  one  or  both 
planes  of  projection,  the  above  problem  is  inserted  here  among 
roblems  of  oblique  projections. 


80  PROJECTIONS  OP   SOLIDS. 

liCt  VB  be  the  radius  of  the  circle,  which,  with  the  pciint  V,  is 
the  horizontal  projection  of  tlie  cone.  Since  the  base  of  the  cone 
rests  in  tlie  horizontal  plane  of  projection,  C'B'  is  its  vertical 
projection.  Since  the  axis  of  the  cone  is  vertical,  V,  the  vertica' 
projection  of  the  vertex,  must  be  in  a  perpendicular  to  the  ground 
line,  through  V,  and  may  be  assumed,  unless  the  height  of  tho 
cone  is  given.  V'C  and  VB',  the  extreme  elements,  as  seen  in 
elevation,  are  parallel  to  the  vertical  plane  of  projection,  hence 
their  horizontal  projections  are  CV  and  BV,  parallel  to  the  ground 
line  (8  e).  Let  it  be  required  to  find  the  horizontal  projection  of 
any  element,  whose  vertical  projection,  V'D',  is  given.  V  is  the 
horizontal  projection  of  V,  and  D',  being  in  the  circumference  ol 
the  base,  is  horizontally  projected  at  D,  therefore  VD  is  the  hori- 
zontal projection  of  that  element  on  the  front  of  the  cone,  whose 
vertical  projection  is  V'D'.  V'D'  is  also  the  vertical  projection  of 
an  element  behind  VD,  on  the  back  of  the  cone.  Having  given, 
VA,  the  horizontal  projection  of  an  element  of  the  cone,  let  it  be 
required  to  iind  its  vertical  projection.  V  is  the  vertical  projec- 
tion of  V,  and  A,  being  in  the  circumference  of  the  base,  is  verti- 
cally projected  at  A'.  Therefore  V'A'  is  the  required  vertical  pro- 
jection of  the  ju'oposcd  line.  In  inking  the  fignre,  no  part  of  the 
plan  i^  heavy  lined,  and  in  the  elevation,  only  the  element  VB'  is 
slightly  heavy. 

Examples. — 1st.  Construct  three  projections  of  a  cone  placed  as 
the  cylinder  is  in  Prob.  13.  « 

2d.  As  the  cylinder  is  in  Prob.  18. 

57.  Pkob.  20.  To  construct  the  projections  of  a  right  hexagonnl 
j^rism  /  whose  axis  is  oblique  to  the  horizontal  plane,  and  parallel 
to  the  vertical  plane.     PI.  III.,  Figs.  33,  34, 

\st.  Commence  by  constructing  the  projections  of  the  same  prism 
as  seen  when  standing  vertically,  as  in  Fig.  33.  The  plan  only  ia 
strictly  needed,  but  the  elevation  may  as  well  be  added  here,  for 
completeness'  sake,  and  because  some  use  can  be  made  of  it. 

2nd.  Draw  J'G",  making  any  convenient  angle  with  the  ground 
line,  and  set  off  upon  it  spaces  equal  to  G'J',  J'H',  and  J'l',  from 
Fig.  33. 

Zrd.  Since  the  i)rism  is  a  right  one,  at  J",  &c.,  draw  perpen- 
diculars to  J"G',  make  each  of  them  equal  to  J'C,  Fig.  33,  and 
draw  F'C,  which  will  be  parallel  to  J'G",  and  will  complete  the 
second  elevation. 

Ath.  Let  us  suppose  that  the  prism  was  moved  from  its  first 


PUOJECTIONS,  OP   SOLIDS.  21 

position,  Fig.  33,  parallel  to  the  vertical  plane,  and  towards  tha 
right,  and  then  inclined,  as  described,  with  the  corner,  C.T',  of  the 
base,  remaining  in  the  liorizontal  plane.  It  is  clear  that  all  points 
of  the  new  plan,  as  B"',  would  be  in  parallels,  as  BB"',  to  the 
ground  line,  through  the  primitive  plans,  as  B,  of  the  same  points. 
It  is  equally  true  that  the  points  of  the  new  plan  will  be  in  perpen- 
diculai's  to  the  ground  line  through  the  new  elevations  B",  <fec.,  of 
ihe  same  points  (15),  hence  these  points  B'",  &c.,  will  be  at  th 
intersections  of  these  two  groups  of  lines.  Thus,  A'"  is  at  the 
intersection  of  AA'"  with  A"A"' ;  C"  is  at  the  intersection  of  OC" 
with  C'C";  K'"  is  at  the  intersection  of  DK'"  with  H"K"',  &c. 

5th.  B"'C"',  F"'E"',  and  G'"K'",  being  the  projections  of  linea 
of  the  prism  which  are  parallel  in  space,  are  themselves  parallel.  A 
similar  remark  applies  to  C"'D'",  A"'F"',  and  H"'G"'.  Observe, 
that  as  the  upper  or  visible  base  is  viewed  obliquely,  it  is  not  seen 
in  its  true  size,  F"'C"' being  less  than  FC,  Fig.  33;  so  that  this 
base  A"'C'",  E'",  does  not  appear  in  the  new  plan  as  a  regular 
hexagon. 

58.  Prob.  21.  To  construct  the  jwojeetions  of  the  prisrn,  given 
in  the  ^^^evious  problem^  when  its  edges  are  inclined  to  both 
2ylanes  of  projection.    PI.  III.,  Fig.  34a. 

If  the  prism,  PL  III.,  Fig.  34,  he  moved  to  any  new  position, 
such  that  the  inclination  of  its  edges  to  the  vertical  plane,  only, 
shall  be  changed,  the  inclination  of  its  edges  to  the  horizontal 
plane  of  projection  being  unchanged,  the  new  plan  will  be  merely 
a  copy  of  the  second  plan,  placed  in  a  new  position.  Let  the  par- 
ticular position  chosen  be  such  that  the  axis  of  the  prism  shall  be 
in  a  plane  perpendicular  to  the  ground  line,  i.e.  to  both  planes  of 
projection ;  then  the  axis  of  symmetry,  C"'G"',  of  the  second 
plan,  will  take  the  position  C""G'"',  and  on  each  side  of  this  line  the 
plan.  Fig.  34  «,  will  be  made,  similar  to  the  halves  of  the  plan  in 
Fig.  34. 

As  the  prism  is  turned  horizontally  about  the  corner  J",  and 
then  transferred,  producing  the  result  that  the  inclination  of  ita 
axis  to  the  horizontal  plane  is  unchanged,  all  points  of  the  third 
elevation,  as  A'"",  C'"",  &c.,  will  be  in  parallels  to  the  ground  line 
through  A",  C",  &c.,  and  in  perpendiculars  to  the  ground  line, 
through  A"",  C"",  &c. 

By  examination  of  this  solution,  and  by  inspection  of  Figs.  34 
and  34a,  it  appears  that  a  change  in  the  position  of  the  axis, 
with  reference  to  but  one  plane  of  projection  at  a  time,  can  be 


22  PROJECTIOXS    OF    SOLIDS. 

represented  directly  from  projections  already  fjiven;  also  that  a 
curve,  beginning  with  the  lir.st  plan,  and  traced  through  the  six 
Hgures  composing  the  three  given  pairs  of  projections  in  the  ordcf 
in  whicli  they  must  be  made,  would  be  an  S  curve,  ending  iu  the 
third  elevation. 

59.  Execution. — The  full  explanation  of  the  location  of  the  heavy 
ines  cannot  here  be  given.  The  careful  inquirer  may  be  able  to 
satisfy  himself  that  the  heavy  liues  of  the  h"gures,  as  shown,  are  the 
jirojections  of  those  edges  of  the  prism  which  divide  its  illuminated 
from  its  dark  surfaces. 

00.  Pkob.  22.  To  construct  the  2>'>'ojections  of  a  regular  hexof 
go7ial  2iy'>'aniid^  whose  axis  is  inclined  to  the  horizontal  plane  only. 
PI.  III.,  Figs.  35,  36. 

\st.  Commence,  as  with  the  prism  in  the  last  problem,  by  repre- 
senting the  pyramid  as  having  its  axis  vertical. 

Ind.  Draw  a"d'\  equal  to  a'd\  and  divided  in  the  same  way.  At 
w",  the  middle  point  of  a'V?",  draw  ?i"V"  perpendicular  to  «"«', 
and  make  it  equal  to  w'V,  v^'hich  gives  V"  the  new  elevation  of  the 
vertex.  Join  V"  with  a",  i",  c",  and  d'\  and  the  new  elevation 
will  be  completed. 

3rc?.  Supposing  the  same  translation  and  rotation  to  occur  to  the 
primitive  position  of  the  pyramid,  that  was  made  in  the  case  of  the 
prism  (57,  4?/t),  the  points  of  the  ^le\v  plan,  Fig.  36,  will  be  found 
in  a  manner  similar  to  that  shown  in  Fig.  34.  V"  is  at  the  inter- 
section of  VV"  with  V'V";  c'"  is  at  the  intersection  cc"  with 
c"c"';  d'"  is  at  the  intersection  ofdd'"  with  d"d"\  <fec. 

\th.  The  points,  a"'b"'c"'  ....  /'",  of  the  base,  are  connected 
with  V",  the  new  horizontal  projection  of  the  vertex,  to  com])letc 
the  new  plan.  If  the  pyramid  were  less  inclined,  the  perpendicular 
VV"  would  full  within  the  base,  and  the  whole  base  would  then 
be  visible  in  the  plan.  As  it  is, /'"a"'  and  a"'b"'  are  hidden,  and 
therefore  dotted. 

5th.  The  heavy  lines  are  correctly  placed  in  the  diagram;  also 
the  partially  heavy  lines,  which  are  all  between  Y"'d"'  and  the 
ground  line,  but  the  reasons  for  their  location  cannot  here  be  given, 
beyond  the  general  princi['le  (18-20)  already  given. 

61.  Pjioi!.  23.  To  construct  the  projections  of  the  regular  hexa- 
gonal pyramid,  when  its  axis  is  oblique  to  both  planes  of  projec- 
tion.    PI.  III.,  Fig.  36a. 

Suppose  the  pyramid  here  shown  to  be  the  one  represented  ir 


ELEMENTARY   INTERSECTIONS. 


ficfurcs  35  and  36,  and  suppose  that  it  has  been  turned  about  any 
vertical  line  as  an  axis.  Than,  Jirst,  every  point  of  it  will  move 
horizoniallr/ y  second,  every  point  A'ill  hence  remain  at  the  samelieighi 
as  before ;  third,  therefore,  the  inclination  of  all  the  edges  to  t/t6 
horizontal  plane  will  be  unchanged ;  and  hence,  fourtli,  the  now 
plan,  Fig.  3G«,  will  be  only  a  copy  of  the  second  plan.  Fig.  8G, 
placed  so  that  its  axis  of  symmetry,  N""d"'\  shall  make  any  as- 
sumed angle  with  the  ground  line. 

]5y  [second)  and  (15)  the  points,  as  V"",  of  the  third  elevation, 
will  bo  at  the  intersection  of  parallels  to  the  ground  line,  through 
tlie  corresponding  points,  as  V",  of  the  second  elevation,  with  per- 
pendiculars through  the  same  points,  as  V"'',  seen  in  the  third  plan. 
Observe  that  the  two  points  vertically  projected  in  c",  being  at  the 
same  height  above  the  ground  line,  will  appear  in  the  third  eleva- 
tion at  c'""  and  e'"",  in  the  same  straight  line,  through  c",  and  par- 
allel to  the  ground  line.  (32). 

Remembering  also  that  lines  which  are  parallel  in  space  must 
.have  parallel  projections,  on  the  same  plane,  c""'d""'  will  be  paral- 
lel iof""'a""',  &c.     The  heavy  lines  ai-e  indicated  in  the  figure. 

Example.— Construct  Fig.  36a  from  Fig.  36,  without  a  nev) plan, 
by  taking  a  new  vertical  plane  with  its  ground  line  parallel  to 
Y""d"",  and  revolving  it  directly  back  as  mentioned  in  (49). 

§  IV. —  Special  Elementary  Intersections  and  Developments. 

G2.  The  positions  of  other  planes,  than  those  of  piojection,  are 
indicated  by  their  intersections  with  the  planes  of  projection. 
These  intei'sections  ai'e  called  traces. 

A  plane  can  cut  a  straight  line  in  only  one  point ;  hence,  if  a 
plane  cuts  the  ground  line  at  a  certain  point,  its  traces,  both  being 
in  the  plane,  must  meet  in  that  point. 

In  PI.  I.,  Fig.  5,  hV>h'k'  is  a  plane  p>erpendicidar  to  the  ground 
line,  MG,  and,  therefore,  to  both  planes  of  projection,  and  wo  sed 
that  its  two  traces,  hk'  and  h'h',  are  perpendicular  to  the  ground 
line  at  ^'.  Likewise  in  PI.  I.,  Fig.  15,  kaa't  is  a  plane  perpendicu- 
lar to  the  ground  line  MQ,  and  its  traces  at  and  at  are  perpendi- 
cular to  MQ.  That  is:  if  a  plane  is  perpendicular  to  the  ground 
line,  its  traces  icill  also  he  p)crpendicidar  to  that  line. 

This  is  seen  in  regular  projection,  in  PI.  I,,  Fig.  10,  where  PQ  ia 
the  horizontal  trace,  and  P'Q,  the  vertical  tiace,  of  such  a  plane. 

In  PI.  I.,  Fig.  5,  ¥\\fk  is  ^ plane,  parallel  to  the  vertical  plane, 
and  it  has  only  a  horizontal  trace,  fk,  which  is  parallel  to  thi 
ground  line.     The   same  is  true  for  all   such    planes.     Likewise, 


'lA  ELEME>TARY    IXTERSECTIONS. 

ABa'b'  is  a  horizontal  plane.  All  such  planes  have  only  a  I'criicai 
trace,  as  a'b\  parallel  to  the  growid  line. 

In  PI.  I.,  Fig.  2,  the  plane  Yv'd'd  is  perpendicular  only  to  the 
vertical  plane,  and,  as  the  figure  shows,  tlie  horizontal  trace  only^ 
as  dd\  of  such  a  plane,  is  perpendicular  to  the  ground  line.  Also 
the  angle  v'd'b\  between  the  vertical  trace,  v  d\  and  the  gromid 
line,  is  the  angle  made  by  the  plane  with  the  horizontal  plane. 

In  like  manner,  it  can  easily  be  seen  that,  if  a  plane  be  perpen 
dicular  only  to  the  horizontal  plane,  as  in  case  of  a  partly  open  door, 
its  vertical  trace  only  (the  edge  of  the  door  at  the  hinges)  will  1)6 
perpendicidar  to  the  ground  line,  and  the  angle  between  its  hori- 
zontal trace  and  the  ground  line,  will  be  tlie  angle  nude  by  the 
plane  with  the  vertical  plane  of  projection. 

Finally,  if  a  plane  is  ohllque  to  both  planes  of  projection,  both 
of  its  traces  will  be  oblique  to  the  ground  line,  and  at  the  same 
point.  Thus,  PI.  I.,  Fig.  6,  may  represent  such  a  plane,  having  LF 
for  its  horizontal,  and  L'F  for  its  vertical  trace. 

All  the  principles  just  stated  can  be  simply  illustrated  by  taking 
a  book,  half  open,  for  the  planes  of  projection,  and  either  of  the 
triangles  for  the  given  movable  ]ilane ;  and  when  clearly  under- 
stood, the  following  problems  can  also  be  easily  comprehended. 

Pkob.  24. — To  find  the  curve  of  intersection  of  a  cylinder  with 
a  plane.    PI.  R'.,  Fig.  1. 

Let  the  cylinder,  ADBG — A'B",  be  vertical,  and  the  cutting 
plane,  PQP',  be  per})endicular  only  to  the  vertical  plane.  All 
points  in  such  a  plane  must  have  their  vertical  projections  (that  is, 
must  be  vertically  projected)  in  the  vertical  trace,  QP',  of  the 
plane,  but  the  required  curve  must  also  be  embraced  by  the  visible 
limits,  A'A"  and  B'B",  of  the  cylinder.  Hence,  a'b'  is  the  verti- 
cal projection  of  this  curve.  '  Again,  as  the  cylinder  is  vertical,  all 
points  on  its  convex  surface  must  be  horizontally  projected  in 
ADBG.  Hence,  this  circle  is  the  horizontal  projection  of  the 
required  cuive. 

Prob.  25. —  To  revolve  the  curve  found  in  the  last  p>roblem,  so  as 
to  show  its  true  size. 

"When  a  plane  revolves  about  any  line  in  it  as  an  axis,  every 
point  of  it,  not  in  the  axis,  moves  in  a  circular  arc,  whose  radii  are 
all  jjcrpendicular  to  the  axis.  Tlie  representation  of  the  revolutioc 
is  mucli  simplified  by  taking  the  axis  in,  parallel  to,  or  j^erpendicu 
lar  to,  a  plane  of  projection  (31). 


pi-.in. 


K^ 


I 


c 


ELEMENTARY    INTERSECTIONS.  25 

Let  AB — a'J>\  tlie  longer  axis  of  the  curve,  and  which  is  parallel 
to  the  vertical  plane  of  projection,  be  taken  as  the  axis  of  revolu- 
tion. The  curve  may  then  be  revolved  till  parallel  to  that  plane, 
when  its  real  size  and  form  will  appear.  Then,  nt  c',  <:?',  &c.,  the 
vertical  projections  of  C  and  H,_D  and  G,  &c.,  draw  perpendicu- 
Jars,  as  c"A",  to  a'h\  and  make  c'c" ~c'h" —nC  Proceed  likewis 
at  d\  &c.,  since  the  lines,  as  nC,  being  parallel  to  the  horizonta, 
plane,  are  seen  in  their  true  size  in  horizontal  projection  ;  and  join 
the  points  a'h"(j'\  tfcc.,  which  will  give  the  required  true  form  and 
size  of  the  curve  of  intersection  before  found. 

Example.— This  curve  is  an  oval,  called  an  ellipse.  Its  true  size 
could  have  been  shown  by  revolving  its  original  position  about 
DG  as  an  axis,  till  parallel  to  the  horizontal  plane.  The  student 
may  add  this  construction  to  the  plate. 

Prob.  2G. — To  dcvclope  the  portion  of  the  cylinder,  PI.  IV., 
Fig.  1,  below  the  cutting  plane^  PQP'. 

The  convex  surface  of  a  cylinder  is  wholly  composed  of  straight 
lines,  called  elements,  parallel  to  its  axis.  The  convex  surface  of  a 
cone  is  composed  of  similar  elements,  all  of  which  meet  at  its  vor- 
tex. Hence,  each  of  these  surfaces  can  evidently  be  rolled  upon  a 
plane,  till  the  element  first  placed  in  contact  with  the  plane,  returns 
into  it  again.  The  figure,  thus  rolled  over  on  the  plane,  is  called 
the  development  of  the  given  convex  surface,  and  its  area  equals 
the  area  of  that  surface. 

Suppose  the  cylinder  to  be  hollow  as  if  made  of  tin,  and  to  be 
cut  open  along  the  element  B'5'.  Then  suppose  the  element  A'a' 
to  be  placed  on  the  paper,  as  at  A'a',  Fig.  2,  and  let  each  half  be 
rolled  out  upon  the  paper.  The  part  ADB  will  appear  to  the  left 
of  A'a',  and  the  part  AGB,  to  the  right.  The  base  being  a  circle, 
perpendicular  to  the  elements,  will  develope  into  a  straight  line 
B  B",  Fig.  2,  found  by  making  A'c  =  AC,  Fig.  1,  cd—CT>,  Fig.  1, 
&c.,  and  A7i=AH,  Fig.  1,  &c.  B'B"  may  also,  for  convenience, 
be  A'B',  Fig.  1,  produced.  Then  the  parallels  to  A'«',  through  c 
rf,  &c.,  will  be  developments  of  elements  standing  on  C,  D,  &c^ 
Fig.  1,  and  by  projecting  over  upon  them,  a'  at  a',  c'  at  c'  and  h' ; 

(V  at  d'  and  g\ B'  at  b'  and  b'\  and  joining  the  points,  the 

figure  B'B"6"a'i',  will  be  the  required  development  of  the  cylinder. 
^  Remark. — If,  now,  a  flat  sheet  of  metal  be  cut  to  the  pattern 
just  found,  it  will  roll  up  into  a  cylinder,  cut  olF  obliquely  as  by 
the  plane  PQP'.  By  making  the  angle  P'QA'  of  any  desired  size, 
the  corresponding  flat  pattern  can  be  made  as  now  explained. 


26  ELEMENT AKT  INTEKSECT10N8. 

Prob.  27. —  To  find  the  intersection  of  a  vertical  cone,  icith  a 
plane,  perpendicular  to  the  vertical  2)lane  of  projection.  PL  IV., 
Fig.  3. 

Let  V— ADBC  he  the  plan,  and  A'B'V  the  elevation  of  tbe 
cone,  and  PQ  and  P'Q  the  traces  of  the  given  cutting  plane;  whose 
horizontal  trace,  PQ,  shows  it  (G2)  to  be  perpendicular  to  the  ver- 
tical plane.  For  the  reasons  given  in  Problem  25,  ci'b'  will  be  the 
vertical  projection  of  the  required  curve.  The  convex  surface  of 
the  cone  not  being  vertical,  the  horizontal  projection  of  the  inter- 
section will  be  a  curve,  whicli  must  be  found  by  constructing  it* 
points  as  follows. 

First.  The  method  by  chmients.  Any  line,  as  VE',  is  the  verti- 
cal projection  of  two  elements  whose  horizontal  projections  are 
VE  andVF  (Prob.  19).  Therefore  e',  where  it  crosses  the  vertical 
projection,  a'b\  of  the  intersection,  is  the  vertical  projection  of  t\vo 
points  of  the  required  curve.  Their  horizontal  prp-ections,  e  ami/', 
are  found  by  projecting  e'  down  upon  VE  and  VF.  Other  points 
can  be  found  in  the  same  manner,  except  d  and  //,  since  the  ]»ro- 
jectiug  line  d'd  coincides  with  the  elements  VD  and  VG.  The 
horizontal  projections  of  a'  and  b'  are  a  and  b. 

Second.  The  method  by  circular  sections.  Let  JNFN'  be  the  ver- 
tical trace  of  a  horizontal  auxiliary  plane  through  d' .  This  plane 
will  cut  from  the  cone  the  circle  in'n' — dnxg,  on  which  d'  can  be 
projected  at  d  and  </,  the  points  desired.  Other  points  of  the 
h(jrizontal  projection 'can  be  found  in  the  same  manner. 

Remarks. — a.  The  curve  adbg — a'b'  is  an  ellipse  whose  longer 
axis  is  the  line  ah — a'b',  whose  true  length  is  a'b'.  Its  shorter 
axis  is  the  line  jjq — •/?',  whose  trtic  length  pq  bisects  ah,  and  is 
always  less  than  ah;  since  it  is  a  chord  of  the  circle  x'y'  through 
J)' ,  and  x'lj'  is  easily  seen  to  be  equal  to  ah.  An  ellipse,  having 
thus  two  axes  of  symmetry,  can  be  drawn  by  using  an  arc  of  the 
irregular  curve  that  will  fit  one  ouartcr  of  it. 

h.  On  the  cylinder,  d',  the  middle  of  a'b'  is  on  the  axis  0 — dd', 
That  is,  the  centre  of  the  ellipse  cut  from  a  cylinder,  is  on  the  axis 
of  the  cylinder.  Not  so,  however,  with  the  cone;  p',  the  middle 
o{  a'b' ,  is  not  on  VD',  the  vertical  projection  of  the  axis,  but  is  on 
the  hide  oi' it  towards  the  lowest  ])oint,  bb' ,  of  the  curve  of  inter- 
Bcction.  On  account  of  the  acuteness  of  the  intersections  ^X,  p  and 
q,  these  points  can  better  be  found  as  were  d  and  g. 

Examples. — \st.  To  make  the  horizontal  projection  less  circulai 
than  in  the  figure,  let  the  cone  be  <juite  flat,  as  at  A  VB,  Fig.  0,  and 
with  a'  near  the  vertex,  and  V  quite  near  the  base. 


KLEMENTART   ENTEKSECTTONS.  2T 

2(7.  Find  tliG  true  size  of  the  curve  by  either  of  tlie  ways  indi- 
catecl  in  Prob.  25,  nl.so  by  revolving  the  plane  PQP',  containing 
it,  either,  about  PQ  as  an  axis,  into  the  horizontal  plane;  or, 
about  P'Q  as  an  axis,  into  the  vertical  plane.  In  the  former  case 
it  is  only  to  be  i-emembered  that  e'Q,  for  example,  shows  the  trua 
distance  of  ee'  from  PQ ;  and,  in  the  latter  case,  that  eh,  for  exam 
pie,  shows  the  true  distance  of  ee'  from  the  vertical  trace  P'Q  (6), 

Prob.  28. —  To  develope  the  convex  surface  of  a  cone,  PI.  IV., 
Pig.  4,  together  ivith  ike  curve  of  intersection,  found  in  tlie  last 
problem. 

First.  If  the  element  VB — ^V'B'  be  placed  in  contact  with  the 
paper  at  VB',  and  if  the  cone  be  then  rolled  upon  the  paper  till 
this  element  returns  into  it  again,  as  at  VB",  the  development, 
VB'B",  will  be  made.  As  all  the  elements  are  equal,  and  as  the 
vertex  is  stationary,  the  develojmient  of  the  base  will  be  the  arc 
B'B",  with  a  radius  equal  to  VB',  the  cone's  slant  height  and  of  a 
length  equal  to  the  circumference  ADBG.  Tliis  length  is  found, 
as  in  case  of  the  cylinder,  by  taking  equal  arcs  of  the  base,  so 
small  that  their  chords  shall  be  sensibly  equal  to  them,  and  laying 
off  those  chords  from  B',  on  the  arc  B'B",  till  B"  is  located. 
Thus,  BE  being  one  eighth  of  ADBG,  its  length  is  laid  off  as  at 
B'e"  eight  times  to  find  B". 

Second.  To  show  the  curve,  adbg — a'b\  on  the  development, 
consider  that  only  the  extreme  elements,  as  VB — VB',  show  their 
true  length  in  projection.  Hence,  the  points  between  a'  and  b' 
must  be  revolved  aroimd  the  axis  of  the  cone,  into  these  elements, 
in  order  to  show  their  true  distances  from  the  vertex.  This  axis 
being  vertical,  the  arcs  of  revolution  will  be  horizontal,  and  will 
therefore  be  vertically  projected  in  the  horizontal  lines  c't<,  d'ti,  &c., 
and  Yu,  Yn,  &c.,  will  be  the  true  distances  of  c',  d' ,  &c.,  from  the 
vertex.  Hence,  make  Y'a"  —  Y'a' ;  Y'u',  and  Y'n"=Y'u;  Y'n',  and 
Vn"  =Yn,  &c.,  and  the  curve  b'a"b''  wtU  be  the  development  of 
the  intersection  of  the  plane  PQP'  with  the  cone. 

Remark, — The  remarks  made  upon  the  development  of  the 
cylinder  equally  apply  here. 

Prob.  29. — To  find  the  intersection  of  a  verticcd  cylinder  with 
two  horizontal  ones  ;  their  axes  being  in  a  plane  parallel  to  the  ver- 
tical p)lane  of  lyrojecti  on.     PI.    IV.,  Pig.  5. 

ABE— A'BA"B"  is  the  vertical  cylinder,  and  MNQR— O'P'O' 
P'  the  lower  horizontal  cylinder. 


28  BLEMENTAEY    INTEUSECTIOXS. 

First.  To  find  the  highest  and  lowest,  and  foremost  and  hind 
most  points  of  the  intersection.  Since  the  horizontal  cylinder  is 
the  smaller  one,  it  will  enter  the  vertical  cylinder  on  one  side,  and 
leave  it  on  the  other,  giving  two  curves ;  but  as  one  cylinder  is  ver- 
tical, and  the  intersection,  being  common  to  both,  is  on  it,  the 
horizontal  piojections  of  both  curves  are  known  at  once  to  be 
CAE  and  DBF.  Now  A  is  the  horizontal  projection  of  both  the 
highest  and  lowest  points  of  CAE.  Tiieir  vertical  projections  are 
a"  and  a'.  Also  C  and  E  are  the  horizontal  projections  of  the  fore- 
most and  hindmost  points,  and  c',  on  M'N',  midway  between  O 
and  O",  is  the  vertical  projection  of  both  of  them.     (41.) 

In  like  manner  h\  b"  and  d'  are  found. 

Second.  To  find  other  intermediate  points.  Take  the  two  points 
whose  horizontal  projection  is  G,  for  examjjle.  They  are  on  the 
horizontal  elements,  one  on  the  upper,  and  the  other  on  the  lower 
half  of  the  horizontal  cylindei",  and  whose  horizontal  projection  is 
ST.  But  to  find  their  vertical  projections,  we  must  revolve  one  of 
the  bases,  as  MQ,  till  })arallel  to  a  plane  of  projection.  Let  this 
base  revolve  about  its  vertical  diameter,  O — O'O",  till  paiallel  to 
the  vertical  plane,  when  OM" — 0'M"'0"  will  be  tiie  vertical  pro- 
jection of  its  front  half  In  this  revolution  the  points,  S,  revolve 
to  S",  and  will  thence  be  vertically  projected  at  XT'  and  S'.  In 
counter  revolution,  these  points  return  in  liorizontal  arcs  to  u'  and 
%\  and  icv'  and  s't'  are  the  vertical  projections  of  the  elements  ST. 
Hence,  project  G,  and  also  H,  at  g'  and  g%  K  and  A",  and  we  shall 
have  foui'  more  points  of  intersection.  Any  numbei"  of  points  can 
be  similarly  found. 

Examples. — \st.  The  last  four  points  could  as  easily  have  been 
found  by  revolving  the  base  MQ  about  the  horizontal  diameter 
MQ — M',  till  parallel  to  the  horizontal  plane.  This  construction 
is  left  for  the  student. 

Id.  If  the  axes  did  not  intersect  each  other,  as  at  IF,  the  points 
C  and  E  would  not  be  equidistant  from  OP,  and  would  not  have 
one  point,  c\  for  their  vertical  projection,  and  the  vertical  projec- 
tion of  the  back  half,  as  AE,  of  each  curve  would  be  a  dotted  line, 
separate  from  the  same  projection  of  the  front  half  The  student 
may  construct  this  case,  also  that  where  one  of  the  elements,  MN 
or  QR,  does  not  intersect  the  vertical  cylinder. 

Zd.  The  horizontal  cylinder,  Fig.  6,  shows  that  when  the  two 
cylinders,  placed  as  before,  are  of  equal  diameter,  the  vertical  pro- 
jections of  their  curves  of  intersection  are  straight  lines.  Hence, 
each  of  the  curves  tlicmselvcs  is  contained  in  a  plane,  that  is,  it  is 


ELBMEXTAEY    INTERSECTIONS.  2^ 

a  *^ plane  ciirvey  This  figure,  if  regarded  sepnrately,  as  a  plan 
view,  therefore  may  represent  the  plan  of  the  intersection  of  two 
equal  semi-circular  arches,  and  the  curves,  IvL  and  AY,  of  inter- 
section, will  be  ellipses. 

The  curves  on  the  cylinders  in  Fig.  5  cannot  be  contained  in 
planes.     Such  curves  are  said  to  be  of  double  curvature. 

4th.  By  developing  the  cylinders,  in  Figs.  4  and  5,  as  in  Fig.  2, 
the  patterns  may  be  found  which  will  give  intersecting  sheet  metal 
pipes,  when  rolled  up  in  cylindrical  form.  The  student  should 
consti'uct  these  developments,  also  the  case  in  which  the  vertical 
cyUnder  should  be  the  smaller  one. 


Pkob.  30. —  To  find  the  intersection  of  a  horizontal  cylinder  with 
a  vertical  cone.     PI.  IV.,  Fig.  7. 

Let  ABV  be  the  vertical  projection  of  a  cone,  and  let  the  circle 
with  radius  o«,  be  an  end  view  of  the  cylinder ;  its  axis,  o'o",  in- 
tersecting A'V',  that  of  the  cone.  Let  PQ  be  the  vertical  trace 
of  a  second  vertical  plane,  perpendicular  to  the  ground  line,  as 
in  PL  I.,  Figs.  15  and  16,  andlet  V'E'D'be  the  vertical  projection 
of  the  cone,  and  Qi'Qi"WW  that  of  the  cylinder,  on  this  plane. 
In  this  construction,  therefore,  two  vertical  projections  are  em- 
ployed, instead  of  a  horizontal  and  vertical  projection,  for  any  two 
projections  of  an  object  are  enough  to  show  its  form  and  position. 
This  will  more  readily  appear  by  tui-ning  the  plate  to  bring  VAB 
below  PQ,  when  PQ  will  be  the  ground  line,  the  right  hand  pro- 
jection a  plan,  and  the  left  hand  one  an  elevation,  like  Fig.  5. 

Now  to  find  the  intersection.  Speaking  as  if  facing  the  vertical 
plane  of  projection,  represented  by  the  paper  to  the  left  of  PQ, 
after  revolving  that  plane  about  PQ  into  the  paper,  AV — A'V  is 
the  foremost  element,  and  a'  is  found  by  projecting  a  across  upon 
A'V.  Next,  DV  is  the  right  hand  projection  of  two  elements, 
whose  left  hand  i)rojections  are  E'V  and  \)'\' .  We  therefore 
project  G  at//'  and  e'. 

To  find  intermediate  points.  Assume  any  element  FV,  draw 
FF"  perpendicular  to  AB,  then  make  an  arc,  AF",  of  the  plan  of 
the  cone's  base,  and  make  A'F'  =  ATI' =  FF".  Then  VF'  and 
VH'  will  be  the  left  hand  projections  of  the  two  elements  project- 
ed in  FV.  Then  project/*  at /"and  h'  on  these  elements,  andff'a'h'e' 
will  be  the  visible  part  of  the  intersection.  Its  right  hand  projec- 
tion is  afG,  where /"and  G  are,  each,  the  pi'ojection  of  two  poiats 
on  opposite  sides  of  the  cone. 


so  KLEJkrENTARY   INTEKSECTIONS. 

Example. — By  (levcloping  tlie  cone  and  the  cylinder,  pattorm 
could  be  made  for  a  conical  pipe  entering  a  cylindrical  one. 

63.  Observing  that,  in  every  case,  the  awxiliary  planes  are 
made  to  cut  the  given  curved  surfaces  in  the  simplest  manner,  that 
is,  in  straight  lines  or  circles,  we  have  the  following  ]iriiKipli'S.  To 
cut  right  lines,  at  once,  from  two  cylinders^  as  in  Fig.  5,  a.  j^hme 
*nust  be  parallel  to  both  their  axes.  To  cut  a  cylinder  and  cone,  at 
once,  in  the  same  manner,  as  in  Fig.  T,  each  plane  must  contain  the 
vertex  of  the  cone,  and  be  parallel  to  the  axis  of  the  cylinder.  To 
out  elements  at  once  fiom  tv>o  cones,  a  plane  must  simply  contain 
both  vertices. 

Examples. — \st.  Thus,  in  Fig.  10,  all  planes  cutting  elements,  both 
from  cone  W,  and  cone  AA',  will  contain  the  line  VAB,  hence 
their  traces  on  the  horizontal  plane  will  merely  pass  through  B. 
Thus  the  plane  BD  cuts  from  the  cone,  V,  the  elements  Y'a' — Va, 
and  W — Yc ;  and  from  the  cone,  A,  the  elements  A'D' — Ad,  and 
A'd' — AD.  The  student  can  complete  the  solution,  the  remainder 
of  which  is  very  similar  to  the  two  preceding. 

2d.  To  find  the  intersection  of  +1  sphere  and  cone,  PI.  IV., 
Fig.  1 1 ,  auxiliary  planes  may  most  conveniently  be  placed  in  two  ways. 
First,  horizontally.  Then  each  will  cut  a  circle  fiom  the  sj)here, 
Efnd  one  from  the  cone  ;  whose  horizontal  projections  will  be  circles, 
and  whose  intersections  will  be  points  of  the  intersection  of  the 
cone  and  sphere.  Second,  vertically.  Then  each  plane  must  con- 
tain the  axis  of  the  cone,  from  which  it  will  cut  two  elements.  It 
will  also  cut  the  sphere  in  a  circle,  and  by  revolving  this  plane 
about  the  axis  of  the  cone  till  parallel  to  the  vertical  plane,  as  in 
Prob.  17,  the  intersection  of  the  circle  with  the  revolved  elements, 
see  Prob.  27,  may  be  noted,  and  then  revolved  back  to  their  true 
position.  The  student  can  readily  make  the  constructicm,  after  due 
familiarity  with  preceding  problems  has  made  the  apprehension  of 
the  present  article  easy. 

Pkob.  31. —  To  find  the  intersection  of  a  vertical  hexagonal 
prism  with  a  sphere,  whose  centre  is  in  the  axis  of  the  prism,  PI. 
IV.,  Fig.  8. 

Let  O — ABC  be  part  of  the  sphere,  and  DGIIK  the  prism, 
showing  one  lace  in  its  real  size,  and  therefore  requiring  no  ])lan 
(47).  Draw  dg  parallel  to  AC,  and  the  arc  eA/'with  O  as  a  centre, 
and  through  e  and/.  Tiiis  arc  is  the  real  size  of  the  intersection 
of  tlic  middle  face  of  the  prism  with  the  surface  of  the  sphere.  AU 
the  faces,  being  equal,  have  circular  tops,  equal  to  ehf ;  but,  being 


ELEMEXTARY    INTERSECTION'S.  31 

seen  obliquely,  they  would  be  really  elliplical  in  ijrojectioii.  It  h 
oi'dinarily  sufficient,  however,  to  represent  them  by  circular  arcs, 
tangent  to  hn^  the  horizontal  tangent  at  4,  and  containing  tiic 
points  d  and  e,  and ^Z"  and  g,  as  shown. 

Hemark. — The  heavy  lines  here,  show  the  ynX  of  the  prism 
within  the  sphere,  as  a  spherical  topped  bolt  h«.-ij.  To  make 
Df?=EF,  draw  Od  at  45°  with  AC,  to  locate  dg.  To  make  the 
sphei'ical  top  flatter,  for  the  same  base  DG,  take  a  iyrgm'  sphere, 
and  a  plane  above  its  centre  for  the  base  of  the  prism. 

Prob.  32. — To  construct  the  intersection  of  a  vertical  cone  tcith 
a  vert  iced  hexagonal  prism  j  both  having  the  same  axis.  PI.  IV., 
Fig.  9. 

Let  YAB  be  the  cone,  and  CFGII,  the  prism,  whose  elevation 
can  be  made  without  a  plan  (48),  since  one  face  is  seen  in  its  real 
size.  The  semicircle  on  cf  is  evidently  equal  to  that  of  the  cir- 
cumscribing circle  of  the  base  of  the  prism,  and  ct  is  the  chord  of 
two  thirds  of  it.  Then  half  of  ct,  laid  ofi'  ou  either  side  of  O,  the 
middle  of  CF,  as  at  0«,  will  give  ?ijo.the  pj-ojection  of  the  middle 
face  EDc?  after  turning  the  prism  90°  about  its  axis.  This  done, 
np  Avill  be  the  height,  above  the  base,  of  the  highe.'it  point  at  which 
this  and  all  the  faces  will  cut  the  cone.  A  vertical  plane,  not 
through  the  vertex  of  a  cone,  cuts  it  in  the  curve,  or  "  conic  sec- 
tion," called  a  hyperbola.  The  vertical  edges  of  the  j^rism  cut  the 
cone  at  the  height  F/,  hence,  drawing  the  curves,  as  dse.,  sharply 
curved  as  at  s,  and  nearly  straight  near  d  and  e,  we  shah  have  a 
sufficiently  exact  construction  of  the  required  intersection. 

Remark. — ^The  heavy  lines  represent  the  part  of  the  prism  within 
the  cone,  finished  as  a  hexagonal  head  to  an  iron  "  bolt,"  such  as  is 
often  seen  in  machinery.  The  horizontal  top,  hg.,  of  the  head,  may 
be  drawn  by  bisecting  pr  at  g.  To  make  Cc=ED,  as  is  usual  in 
practice,  simply  draw  Oc  at  an  angle  of  45°  a\  itli  AB,  to  locate  cf 
By  making  VAB  =  30°  perhaps  the  best  proportions  will  be  found. 

(34.  In  the  subsequent  ajiplications  of  projections  hi  practical 
problems,  the  ground  line  is  very  generally  omitted;  since  a  know- 
ledge of  the  object  represented  makes  it  evident,  on  invspection, 
which  are  the  plans,  and  which  the  elevations. 

General  Examples. 
The  careful  study  of  the  detailed  explanations  of  the  preced- 
ing problems,  will  enable  the  student  to  solve  any  of  the  follow- 
ing additional  examples. 


32  ELEMENTARY    INTERSECTIONS. 

Ex.  1. — In  Prob.  24,  substitute  for  the  cylinder  any  pr  dn, 
find  the  intersection  with  the  plane  PQP',  and,  by  Prob.  25,  -rind 
the  true  form  and  size  of  this  intersection. 

Ex.  2. — In  Prob.  27,  substitute  for  the  cone  any  pyramid. 
Vary  this  and  Ex.  1  by  dillcrcnt  positions  of  PQP',  cutting 
both  hoses  in  tlx.  1. 

Ex.  3.— In  Ex.  2,  find,  by  Prob.  25  or  by  Prob.  27,  Ex.  2d, 
the  true  form  and  size  of  the  intersection  and,  by  Prob.  28,  the 
development  of  tlie  convex  surface  of  the  pyramid. 

Ex.  i. — In  Probs.  22,  23,  substitute  for  the  pyramid  a  cone 
whose  convex  surface,  rolling  on  H  (23),  shall  be  shown,  first, 
with  its  axis  parallel  to  V;  and,  second,  with  its  axis  oblique  to  V. 

Ex.  5. — In  Ex.  4,  find  the  intersection  of  the  cone  with  any 
plane  parallel  to  II;  and  show  the  curve  on  both  positions  of  the 
cone. 

Ex.  C. — In  Ex.  5,  let  the  cutting  plane  be  vertical  but  ob- 
lique to  V,  and  not  containing  the  cone's  vertex. 

Ex.  7. — In  Prob.  20,  let  the  horizontal  cylinder  be  the  larger 
one,  and,  after  finding  its  intersection  with  the  vertical  one,  de- 
velope  it. 

Ex.  8.   In  Probs.  22,  23,  substitute  for  the  pyramid  a  cylinder. 

Ex.  9. — In  Probs.  22,  23,  substitute  for  the  pyramid  a  hollow 
hemisphere. 

Ex.  10. — In  Prob.  29,  let  the  axis  of  the  horizontal  cylinder  be 
inclined  first  to  II  only,  and  then  to  both  H  and  V. 

Ex.  11. — In  Probs.  22,  23,  let  the  pyramid,  when  in  the  posi- 
tion shown  in  Fig.  3G  (but  more  inclined),  rest  its  edge  V"a'" 
against  an  upper  edge  of  a  cube  standing  on  11. 

Ex.  12. — Find  the  four  following  sections  of  a  sphere:  one  by 
a  horizontal  plane,  one  by  a  plane  parallel  to  V,  one  by  a  vertical 
plane  ol)liquc  to  V,  and  one  by  a  plane  perpendicular  to  V  and 
oblique  to  n. 

Ex.   13. — Cut  a  regular  hexagon  from  a  cube. 

Ex.  14. — Cut  a  rhombus  and  an  isosceles  triangle  from  the 
square  prism.     PI.  II.,  Fig.  17. 

Ex.  15. — Construct  the  ])rojections  of  the.  cylinder,  PI.  IV., 
Fig.  1,  after  rotating  it  and  PQP',  together,  45°  on  its  axis. 

Ex.  IC— Sul)stitute  for  the  blocks,  PI.  III.,  Figs.  28,  29,  a 
pile  of  thin  cylinders  of  unequal  diameters,  but  with  a  common 
axis  placed  obliquely  to  V. 


PL.iy 


DIVISION    SECOND. 

DETAILS  OF  MASONRY,  WOOD,  AND  METAL   CONSTRUCTIONS. 


CHAPTER  I. 

CONSTRUCnOXS   IN   MASONRY. 

§  1. — General  Definitions  and  Principles  aj^plicable  both  to  JBrick 
and  &tone-work. 

65.  A  horizontal  layer  of  brick,  or  stone,  is  called  a  course.  The 
seam  between  two  courses  is  called  a  coursing-joint.  The  seam 
between  two  stones  or  bricks  of  the  same  course,  is  a  vertical  or 
heading-joint.  The  vertical  joints  in  any  course  sliould  abut  against 
the  solid  stone  or  brick  of  the  next  courses  above  and  below.  This 
arrangement  is  called  breaking  joints.  The  particular  arrangement 
of  the  pieces  in  a  wall  is  called  its  bond.  As  far  as  possible,  stones 
and  bricks  should  be  laid  with  their  broadest  surfaces  horizontal. 
Bricks  or  stones,  whose  length  is  in  the  direction  of  the  length  of  a 
wall,  are  called  stretchers.  Those  whose  length  is  in  the  direction 
of  the  thickness  of  a  wall,  are  called  headers. 

§  11.— Brick   Work. 

06.  If  it  is  remembered  that  bricks  used  in  building  have,  usually, 

n  invariable  size,  8"  x  4"  X  2"  (the  accents  indicate  inches),  and 

bat  in  all  ordinary  cases  they  are  used  Avhole,  it  will  be  seen  that 

brick  walls  can  only  be  of  certain  thicknesses,  while,  in  the  use  of 

Btone,  the  wall  can  be  made  of  any  thickness. 

Thus,  to  begin  with  the  thinnest  house  wall  which  ever  0C(;ur8, 
viz.  one  whose  thickness  equals  the  lengtii  of  a  brick,  or  8  inches  ; 
the  next  size,  disregarding  for  the  present  the  thickness  of  mortar^ 
would  be  the  length  of  a  brick  added  to  the  width  of  one,  or  e4ua] 
to  the  width  of  three  bricks,  making  12  inches,  a  thickness  empK/yed 
in  the  ]>artition  avails  and  upper  stories  of  first  class  houses,  oi    tbo 


COXSTRL'CTIOXS    IX    MASOXRY, 

outside  walls  of  small  houses.  Then,  a  wall  whose  thickness  ia 
equal  to  the  length  of  two  bricks  or  the  wiilth  of  four,  making  16 
inches,  a  thickness  proper  for  the  outside  walls  of  the  lower  stories 
of  first  class  liouses ;  and  lastly,  a  wall  whose  thickness  equals  the 
length  of  two  bricks  added  to  the  width  of  one;  or,  equals  tlic 
width  of  five  l)ricks,  or  20  inches,  a  thickness  proper  for  the  base- 
ment walls  of  first  class  houses,  for  the  lower  stories  of  few-storied, 
heavy  manufactory  buildings,  &c. 

G7.  In  the  common  bond,  generally  used  in  this  country,  it  may 
be  observed — 

a.  Tliat  in  heavy  buildings  a  common  rule  appears  to  be,  to  have 
one  row  of  headers  in  every  six  or  eight  rows  of  bricks  or  courses, 
i.e.  five  or  seven  rows  of  stretchers  between  each  two  successive 
rows  of  headers  ;  and, 

b.  That  in  the  12  and  20  inch  walls  there  may  conveniently  be  a 
row  of  headers  in  the  back  of  the  wall,  intermediate  between  the 
rows  of  headers  in  the  face  of  the  wall,  while  in  the  8  inch  and  16 
inch  walls,  the  single  row  of  headers  in  the  former  case,  and  the 
double  row  of  headers  in  the  latter,  would  take  up  the  whole  thick- 
ness of  the  wall,  and  there  might  be  no  intermediate  rows  of 
headers. 

c.  The  separate  rows,  making  up  the  thickness  of  the  wall  in  anj 
one  layer  of  stretchers,  are  made  to  break  joints  in  a  liorizontal 
direction,  by  inserting  in  every  second  row  a  half  brick  at  the  end 
of  the  wall. 

68.  Calling  the  preceding  arrangements  connnon  bonds,  let  us 
next  consider  the  bonds  used  in  the  strongest  engineering  works 
which  aie  executed  in  brick.  These  are  the  £Jnglish  bond  and.  the 
Fleynish  bond. 

The  J^nfjUsh  Bond. — In  this  form  of  bond,  every  second  course, 
as  seen  in  the  face  of  the  wall,  is  composed  wholly  of  headers,  the 
intermediate  courses  being  composed  entirely  of  stretchers.  Hence, 
in  any  practical  case,  we  have  given  the  thickness  of  the  wall  and 
the  arrangement  of  the  bricks  in  tlie  front  row  of  each  course,  and 
are  required  to  fill  out  the  thickness  of  the  wall  to  tlie  best  advantage, 

llic  Flemish  Bond. — In  this  bond,  each  single  course  consistfc 
of  alternate  headers  and  stretchers.  The  centre  of  a  header,  in 
any  course,  is  over  the  centre  of  a  stretcher  in  the  course  next 
al)0vc  or  below.  The  face  of  the  wall  being  thus  designed,  it 
remains,  as  before,  to  fill  out  its  thickness  suitably. 

69.  Example  1.  To  represent  an  Eight  Inch  Wall  in  Eng-- 
lish  Bond.    Let  each  course  of  stretchers  consist  of  two  rows,  sidfl 

3 


CONSTRUCTIONS    IN    MASONRY.  35 

bv  side,  the  bricks  in  wliicli,  break  joints  with  each  other  1,  ori- 
zoiitally.  Tlien  the  joints  in  the  courses  of  headers,  will  be  distant 
half  the  width  of  a  brick  from  the  vertical  joints  in  the  adjacent 
courses  of  stretchers,  as  may  be  at  once  seen  on  constructing  a 
diagram. 

VO.  Ex.  2.  To  represent  a  Twelve  Inch  Wall  in  English 
Bond.  Sec  PL  V.,  Fig.  37.  In  the  elevation,  four  courses  are 
§hoAvn.  The  upper  plan  represents  the  topmost  course,  and  in  the 
lower  plan,  the  second,  course  from  the  top  is  shown.  Tlie  courses 
having  stretchers  in  the  foce  of  the  wall,  could  not  be  filled  out  by 
two  additional  rows  of  stretchers,  as  such  an  arrangement  would 
cause  an  unbroken  joint  along  the  line,  «J,  throughout  the  Avhole 
height  of  the  wall — since  the  courses  having  headers  in  the  face, 
must  be  filled  out  with  a  single  row  of  stretchers,  in  order  to  make 
a  twelve  inch  wall,  as  shown  in  the  lower  plan. 

In  order  to  allow  the  headers  of  any  couise  to  break  joints  with 
the  stretchers  of  the  same  coui'se,  the  row  of  headers  may  be  filled 
out  by  a  brick,  and  a  half  brick — split  lengthwise — as  in  the  upper 
plan  ;  or  by  two  three-quarters  of  bricks,  as  seen  in  the  lower 
plan. 

71.  Ex.  3.  To  represent  a  Sixteen  Inch  Wall  in  English 
Bond.  The  simplest  plan,  in  which  the  joints  would  overlap  pro- 
perly, seems  to  be,  to  have  every  second  course  composed  entirely  of 
lieaders,  breaking  joints  horizontally,  and  to  have  the  intermediate 
courses  composed  of  a  single  row  of  stretchers  in  the  front  and 
back,  with  a  row  of  headers  in  the  middle,  which  would  break 
joints  with  the  headers  of  the  first  named  courses.  If  the  stretcher 
courses  were  composed  of  nothing  but  stretchers,  there  would 
evidently  be  an  unbroken  joint  in  the  middle  of  the  wall  extending 
through  its  who-le  height. 

72.  Ex.  4.  To  represent  an  Eight  Inch  Wall  in  Flemish 
Bond.  PI.  v.,  Fig.  38,  shows  an  elevation  of  four  courses,  and  the 
plans  of  two  consecutive  courses.  The  general  arrangement  of  both 
courses  is  the  same,  only  a  brick,  as  AA',  in  one  of  them,  is  set  sis 
inches  to  one  side  of  the  corresponding  brick,  B,  of  the  next  course 
— measuring  from  centre  to  centre. 

73.  Ex.  5.  To  represent  a  Twelve  Inch  Wall  in  Flemish 
Bond.  PI.  v.,  Fig.  39,  is  arranged  in  general  like  the  preceding 
figures,  with  an  elevation,  and  two  plans.  One  course  being  arranged 
as  indicated  by  the  lower  plan,  the  next  course  may  be  made  up  in 
two  ways,  as  shown  in  the  upper  plan,  where  the  grouping  shown 
at  the  right,  obviates  the  use  of  half  bricks  in  every  second  course. 


36 


coxsTRrcrio.vs  in  masoxiiy. 


riiere  seems  to  be  no  other  simple  way  of  combining  ilie  Inicks  ir 
this  wall  so  as  to  avoid  the  use  of  half  bricks,  without  lea\iny  ujicn 
spaces  in  some  parts  of  the  courses. 

74.  Kx.  0.  To  represent  a  Sixteen  Inch  Wall  in  Flemish 
Bond.  PI.  v.,  Fig.  40.  The  ligure  explains  itself  sutHciently. 
IJrieks  may  not  only  be  split  crosswise  and  lengthwise,  but  even 
thicknesswise,  or  so  as  to  give  a  piece  8x4x1  niches  in  size. 
Alihough,  as  has  been  remarked,  whole  bricks  of  the  usual  dhuen- 
sions  can  only  foi-m  walls  of  certain  sizes,  yet,  by  inserting  frag- 
ments, of  jiroper  sizes,  any  length  of  wall,  as  between  windows  and 
doors,  or  width  of  jiilasters  or  panels,  may  be,  and  often  is,  con 
structed.  By  a  similar  aititice,  and  also  by  a  skilful  disposition  of 
the  mortar  in  the  vertical  joints,  tai)ering  structures,  as  tall  cliim- 
neys,  are  formed. 

§  UL— Stone  Work. 

75.  The  following  examples  will  exhibit  the  leading  varieties  of 
arrangement  of  stones  in  walls. 

Example  1.  Regular  Bond  in  Dressed  Stone.  PI.  YL,  Fig.  41. 
Here  the  stones  are  laid  in  regular  courses,  and  so  that  the  middle 
of  a  stone  in  one  course,  abuts  against  a  vertical  joint  in  the  course 
above  and  the  course  below.  In  the  present  example,  those  stones 
whose  ends  appear  in  the  front  face  of  the  wall,  seen  in  elevation, 
take  up  the  whole  thickness  of  the  wall  as  seen  in  plan. 

The  right  hand  end  of  the  wall  is  represented  as  broken  down  in 
all  the  figures  of  this  plate.  Broken  stone  is  represented  by  a 
smooth  broken  line,  and  the  under  edge  of  the  outhanging  part  of 
any  stone,  as  at  w,  is  made  heavy. 

7G.  Ex.  2.  Irregular  Rectangular  Bond.  PI.  YL,  Fig.  4:.\  In 
this  example,  each  stone  has  a  rectangular  face  in  the  front  of  the 
wall.  These  faces  are,  however,  rectangles  of  various  sizes  and 
proportions,  but  arranged  with  their  longest  edges  horizontal,  and 
also  so  as  to  break  joints. 

77.  That  horizontal  line  of  the  ))lan  which  is  nearest  to  the  lower 
border  of  the  plate,  is  evidently  the  plan  of  the  top  line  of  the  elO' 
vation,  hence  all  the  extremities,  as  a',  b\  «fec.,  of  vertical  joints,  found 
on  that  line,  must  be  horizontally  projected  as  at  a  and  ^,  in  the 
horizontal  jji-djection  of"  (he  same  line. 

78.  Ex.  3.  Rubble  Walls.  The  remaining  figures  of  PI.  YL, 
represent  various  forms  of  "  rubble  "  wall.  Fig.  43  represents  a 
wall  of  broken  boulders,  or  loose  stones  of  all  sizes,  such  as  are 
found  abundantly  in  New  England.     Since,  of  course,  such  stones 


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CONSTRUCTIONS    IX    ilASOXRT,  37 

would  not  fit  together  exactly,  the  "chinks"  between  them  are 
filled  with  small  fragments,  as  shown  in  the  figure.  Still  smaller 
irregularities  in  the  joints,  which  are  not  thus  filled,  are  repre- 
sented after  tinting  by  heavy  strokes  in  inking.  Fig.  44  repre- 
sents the  plan  and  elevation  of  a  ruljble  wall  made  of  slate ; 
hence,  in  the  plan,  the  stones  appear  broad,  and  in  the  elevation, 
long  and  thin,  with  chink  stones  of  similar  shape.  Fig.  45 
represents  a  rubble  wall,  built  in  regular  courses,  which  gives 
a  i^leasing  effect,  particularly  if  the  Avail  have  cut  stone  corners, 
of  eqiud  thickness  with  the  rubble  courses. 

Ex.  4.  A  Stone  Box-culvert.  PI.  YL,  Figs.  C,  D, 
E.     Scale  tV  of  an  inch  to  1  ft. 

Fig.  C  is  a  longitudinal  section;  D,  jjart  of  an  end  elevation; 
and  E,  jjart  of  a  transverse  section.  Waste  water  flowing  over 
the  dam  del',  into  the  well  a  between  the  wing- walls  a  and  b'  and 
the  head  h,  escapes  by  the  culvert  cc — c",  which  is  strengthened 
by  an  intermediate  cross- wall  '»i'',  occurring  in  the  course  of  its 
length. 

The  masonry  rests  on  a  flooring  of  2-inch  planks  lying  trans- 
versely on  longitiidinal  sills,  which,  in  turn,  rest  on  transverse  sills. 
Thus  a  firm  continuous  bearing  is  formed  which  prevents  un- 
equal settling  of  the  masonry,  while  washing  out  underneath  is 
provided  against  by  sheet  piling  partly  shown  at  /;,  j/,  p" ,  and 
extending  six  feet  into  the  ground. 

The  student  should  construct  this  example  on  a  larger  scale, 
from  4  to  G  sixteenths  of  an  inch  to  a  foot;  and  should  add  a 
jilan,  or  a  horizontal  section,  both  of  which  may  easily  be  con- 
structed from  the  data  afforded  by  the  given  figures. 

79.  Execution. — Plate  A^I.  may  be,  \st,  pencilled;  2f/,  inked 
in  fine  lines;  Zd,  tinted.  The  rubble  walls,  having  coarser  lines 
for  the  joints,  may  better  be  tinted,  before  lining  the  joints  in 
ink. 

Also,  in  case  of  the  rubble  walls,  sudden  heavy  strokes  may  be 
made  occasionally  iii  the  joints,  to  indicate  slight  irregularities 
in  their  thickness,  as  has  already  been  mentioned. 

The  right  hand  and  lower  side  of  any  stone,  not  joining  an- 
other stone  on  those  sides,  is  inked  heavy,  in  elevation,  and  on 
the  plans  as  usual.     The  left-hand  lines  of  Figs.  43  and  44  are 


38  COXSTRUCTIONS    IX    MASOXRY. 

tangent  at  various  points  to  a  vertical  straight  line,  walls,  such  as 
are  represented  in  those  figures,  being  made  vertical,  at  the  fin- 
ished end,  by  a  plumb  line,  against  "uiiich  the  stones  rest. 

The  shaded  elevations  on  PI.  YI.  may  serve  as  guides  to  the 
depth  of  color  to  be  used  in  tinting  stone  work.  The  tint  actu- 
ally to  be  used,  should  be  very  light,  and  should  consist  of  gi'ay, 
or  a  mixture  of  Ijlack  and  white,  tinged  with  Prussian  blue,  to 
give  a  blue  gray,  and  carmine  also  if  a  purplish  gi-ay  is  desired. 

Remarks. — a.  A  scale  may  be  used,  or  not,  in  making  this 
plate.  The  number  of  stones  shown  m  the  width  of  the  plans, 
shows  that  the  walls  are  quite  thick. 

I.  liubble  walls,  not  of  slate,  are,  strictly,  of  two  kinds:  first, 
those  formed  of  small  boulders,  used  whole,  or  nearly  so  ;  and 
second,  those  built  of  broken  rock.  Each  should  show  the  broad- 
est surfaces  in  ^j)/a«. 

c.  After  tinting,  add  pen-strokes,  called  hatchings,  to  repre- 
sent the  character  of  the  surface;  as  in  Fig.  A.,  for  rough,  or  un- 
dressed stone;  in  waving  rows  from  left  to  right  of  short,  fine, 
equal,  vertical  strokes,  for  smooth  stone;  and  in  a  mixture  of 
numerous  fine  dots  and  small  angular  marks,  for  a  finely  picked- 
up  surface.  (See  actual  stone  work,  good  drafting  cppies,  and 
my  "Drafting  Instruments  and  Operations.") 


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CHAPTER  II. 

OONSTRTTCriONS   IN   WOOD 

§  I. —  General  Memarks. 

80.  Two  or  more  beams  may  be  framed  together,  so  as  to  make 
any  angle  with  each  other,  from  0°  to  180° ;  and  so  tliat  the  plane 
of  two  united  pieces  may  be  vertical,  horizontal,  or  oblique. 

81.  To  make  the  present  graphical  study  of  framings  more  ful'iy 
rational,  it  may  here  he  added,  that  pieces  may  be  framed  with 
reference  to  resisting  forces  which  would  act  to  separate  them  in 
the  direction  of  any  one  of  the  three  dimensions  of  each.  Follow- 
ing out  the  classification  in  the  preceding  article,  let  us  presently 
l)roceed  to  notice  several  examples,  some  mainly  by  general  de- 
scription of  their  material  construction  and  actiofi,  and  some  by  a 
complete  description  of  their  graphical  construction  and  execution^ 
also, 

82.  Two  other  points,  however,  may  here  be  mentioned.  First: 
A  pair  of  pieces  may  be  immediately  framed  into  each  other,  or 
tliey  may  be  intermediately  framed  by  "bolts,"  "keys,"  &c.,  or 
buth  modes  may  be,  and  often  are,  combined.  Second:  Two  com- 
binations of  timbers  which  are  alike  in  general  appearance,  may  be 
adapted,  the  one  to  resist  extension,  and  the  other,  compression, 
and  may  have  slight  corresponding  differences  of  construction, 

83.  JSfote. — For  the  benefit  of  those  who  may  not  have  had  access 
to  the  subject,  the  following  brief  explanation  of  scales,  &c,,  is  here 
inserted,     (See  my  "Drafting  Instruments  and  Operations.") 

Drawings,  showing  the  pieces  as  taken  apart  so  as  to  show  the 
mode  of  union  of  the  pieces  represented,  are  called  '"'•  Details.''^ 

Sections^  are  the  surfaces  exposed  by  cutting  a  body  by  planes, 
and,  strictly,  are  in  the  planes  of  section. 

Sectional  elevations^  or  plans,  show  the  parts  both  ««,  and  be- 
yond, the  planes  of  section. 

Drawings  are  made  in  plan,  side  and  end  elevations,  sections  and 
details,  or  in  as  few  of  these  as  will  show  clearly  all  parts  of  the 
obj'ict  represented. 

84.  In  respect  to  the  instrumental  operations,  these  drawings  are 


40  COXSTRUCTIOXS    IN    -WOOD. 

STijiposed  to  1)0  "  lunde  to  sealc^,"  from  mcasurcment.s  of  models,  or 
fivMii  assii.ned  incnsui'eiiuMits.  It  will,  therefore,  Lc  necessary, 
I)cfore  beginning  the  drawings,  to  explain  the  maimer  of  sketching 
the  oliject,  and  of  taking  and  recording  its  measurements. 

85.  In  sketching  the  object,  make  the  sketches  in  the  same  way 
in  which  th.ey  are  to  be  drawn,  i.e.  in  2^la>^  <^''>'d  elecation,  and  not 
in  pers]iective,  and  make  enough  of  them  to  contain  all  the  mea 
surements,  i.e.  to  show  all  parts  of  the  object. 

In  measuring,  take  measurements  of  all  the  parts  which  are  to  be 
shown ;  and  not  merely  of  individual  parts  alone,  but  such  con- 
necting measurements  as  will  locate  one  pait  with  reference  lo 
another. 

86.  The  usual  mode  of  recording  the  measurements,  is,  to  indi- 
cate, by  arrow  heads,  the  extremities  of  the  line  of  which  the  figures 
between  the  arrow  heads  show  the  length. 

87.  For  brevity,  an  accent  (')  denotes  feet,  and  two  accents  (") 
denote  inches.  The  dimensions  of  small  rectangular  pieces  are 
indicated  as  in  PI.  VII.,  Fig.  50,  and  those  of  small  circular  pieces, 
as  in  Fig.  51. 

88.  In  the  case  of  a  model  of  an  ordinary  house  framing,  such  as 
it  is  useful  to  have  in  the  drawing  room,  and  in  wdiich  the  sill  is 
represented  by  a  piece  whose  section  is  about  2^  inches  by  3  inches, 
a  scale  of  one  inch  to  six  inches  is  convenient.  Let  us  then  describe 
this  scale,  which  may  also  be  called  a  scale  of  two  inches  to  the 
foot. 

The  same  scale  may  also  be  expressed  as  a  scale  of  one  foot  to 
two  inches,  meaning  that  one  foot  on  the  object  is  represented  hy 
two  inches  on  the  drawing  ;  also,  as  a  scale  of  ^,  thus,  a  foot  being 
equal  to  twelve  inches,  12  inches  on  the  object  is  re])resented  by 
two  inclies  on  the  drawing  ;  therefore,  one  inch  on  the  drawing 
represents  six  inches  on  the  object,  or,  each  line  of  the  drawing  is 
\  of  the  same  line,  as  seen  upon  the  object ;  each  line^  for  we  know 
from  Geometry  that  surfaces  arc  to  each  other  as  the  squares  of 
their  homologous  dimensions,  so  that  if  the  length  of  the  lines  of 
tlie  drawing  is  one-sixth  of  the  length  of  the  same  lines  on  the  object, 
the  area  of  the  drawing  would  be  one  thirty-sixth  of  the  area  of  the 
object,  but  the  scale  always  refei's  to  the  relative  lengths  of  tlie 
lines  only. 

89.  In  constructmg  the  scale  above  mentioned,  upon  the  stretched 
drawing  paper,   .see  PI.  VI.,  Fig.  B 

'[St.  Set  off  ujioa  a  tine  straight  pencil  line,  two  inches,  say  thro* 
times,  mal  ing  four  points  of  division. 


CONSTRUCTIONS    IN    WOOO.  41 

2d.  Xumber  the  left  hand  one  of  tliese  points,  12,  the  next,  0, 
,  the  next,  1,  the  next,  2,  &c.,  for  additional  points. 

Sc7:  Since  each  of  tliese  spaces  represents  a  foot,  if  any  one  of 
I  hem,  as  tlie  left  hand  one,  he  divided  into  twelve  equal  parts, 
those  ])arts  will  be  representative  inclies.  Let  the  left  hand  space, 
from  (12)  to  (0)  be  thus  divided,  by  fine  vertical  dashes,  into  twelve 
equal  parts,  making  the  three,  six,  and  nine  inch  marks  longei",  so 
as  to  catch  the  eye,  when  using  the  scale. 

4th.  As  some  of  the  dimensions  of  the  object  to  be  drawn  are 
measured  to  quarter  inches,  divide  the  first  and  sixth  of  the  inches, 
already  found,  into  quarters ;  dividing  two  of  them,  so  that  each 
may  be  a  check  upon  the  other,  and  so  that  there  need  be  no  con 
tinual  use  of  one  of  them,  so  as  to  wear  out  the  scale. 

5th.  When  complete,  the  scale  may  be  inked  ;  the  length  of  it  in 
fine  parallel  lines  about  ^V  of  an  inch  apart. 

90.  It  is  now  to  be  remarked  that  these  spaces  are  always  to  be 
called  by  the  names  of  the  dimensions  they  represent,  and  not 
according  to  their  actual  sizes,  i.  e.  the  space  from  1  to  2  repre- 
sents a  foot  upon  the  object,  and  is  called  a  foot;  so  each  twelfth 
of  the  foot  from  12  to  0  is  called  an  inch,  since  it  represents  an 
inch  on  the  object;  and  so  of  the  quarter  inches. 

91.  Next,  is  to  be  noticed  the  directions  in  which  the  feet  and 
inches  are  to  be  estimated. 

The  feet  are  estimated  from  the  zero  point  towards  the  right, 
and  the  inches  from  the  same  point  towards  the  left. 

Thus,  to  take  off  2' — 5"  from  the  scale,  place  one  leg  of  the 
dividers  at  2,  and  extend  the  other  to  the  fifth  inch  mark  beyond 
0,  to  the  left ;  or,  if  the  scale  Avere  constructed  on  the  edge  of  a 
piece  of  card-board,  the  scale  being  laid  upon  the  paper,  and  with 
its  graduated  edge  against  the  indefinite  straight  line  on  which  the 
given  measurement  is  to  be  laid  off,  place  the  2'  or  the  5"  mark,  at 
that  point  on  the  line,  from  which  the  measurement  is  to  be  laid 
oft',  according  as  the  given  distance  is  to  be  to  the  left  or  right 
of  the  given  point,  and  then  with  a  needle  point  mark  the  5"  point 
or  the  2'  point,  respectively,  which  will,  with  the  given  point, 
include  the  required  distance. 

92.  Other  scales,  constructed  and  divided  as  above  described, 
only  smaller,  are  found  on  the  ivory  scale,  marked  30,  &c.,  mean- 
ing 30  lect  to  the  inch  when  the  tenths  at  the  left  are  taken  as 
feet ;  and  meaning  til ree  feet  to  the  inch  when  tlie  larger  s|)aces 
— three  of  which  make  an  inch — are  called  feet,  and  the  twelfths 
of  the  left  hand  space,  inches.     Intermediate  scales  are  marked 


13  CONSTRUCTIONS    IN   -VTOOIi. 

85,  etc.  Thus,  on  tlie  scale  marked  45,  four  and  a  half  of  the 
hirtrer  spaces  make  one  inch,  and  the  scale  is  therefore  one  of  four 
and  a  lialf  feet  to  one  inch,  wlien  these  spaces  rei)resent  feet ; 
and  of  forty-five  feet  to  one  inch,  when  the  tenths  represent  feet. 
In  like  manner  the  other  scales  may  be  exjjlained. 

So,  on  the  other  side  of  the  ivory,  are  found  scales  marked  |, 
Ac,  meaning  scales  of  -|  inch  to  one  foot,  or  ten  feet,  accoi'ding  as 
tlu!  whole  left  hand  space,  or  its  tenth,  is  assumed  as  lepreseniing 
one  foot.  Kote  that  |  of  an  inch  to  a  foot  is  |  of  a  foot  to  the 
inch,  I  of  an  inch  to  te7i  feet,  is  16  feet  to  an  inch,  &c. 

94.  Of  the  immense  superiority  of  drawing  by  these  scales,  over 
drawing  without  them,  it  is  needless  to  say  much :  without  them, 
we  should  have  to  go  through  a  mental  calculation  to  find  the 
length  of  every  line  of  the  drawing.  Thus,  for  the  piece  which  is 
two  and  a  half  inches  high,  and  drawn  to  a  scale  of  two  inches  to 

a  foot,  we  should  say — 2\  inches  =  j|  of  a  foot^j^  ^^  ^  ^ooX.. 
One  foot  on  the  object =two  inches  on  the  drawing,  then  ^'y  of  a 
foot  on  the  object=2y  of  2  inches=32_  =  J^  of  an  inch,  and  gV  of  a 
foot  (=2|-  inches)  =/y  of  2  inches^/j  of  an  inch. 

A  similar  tedious  calculation  would  have  to  be  gone  through 
with  for  every  dimension  of  the  object,  while,  by  the  use  of  scales, 
like  that  already  described,  we  take  off  the  same  number  of  the  feet 
and  inches  of  the  scale,  that  there  are  of  real  feet  and  inches  in  any 
given  line  of  the  object. 

§  II. — Pairs  of  Timbers  wJiose  axes  make  angles  qfO°  loith  each 

other. 

The  student  should  be  required  to  vary  all  of  the  remaining  con- 
structions in  this  Division,  in  one  or  more  of  the  following  ways. 
First,  by  a  change  of  scale ;  Second,  by  choosing  other  examplet 
from  models  or  otherwise,  but  of  similar  character;  or,  Third,  by 
a  cliange  in  the  number  and  arrangement  of  the  iwojectioris  em- 
ployed  in  representing  the  following  examples. 

95.  Example  1.  A  Compound  Beam  bolted.  PL  VII.,  Fig. 46. 
Medtanical  Construction. 

The  figure  represents  one  beam  as  laid  on  top  of  another.  Thus 
Bituate<l,  the  u])per  one  may  be  slid  upon  the  lower  one  in  the 
direction  of  two  of  its  three  dimensions  ;  or  it  may  rotate  about 
any  one  of  its  three  dimensions  as  an  axis.  A  single  bolt,  passing 
through  both  beams,  as  shown  in  the  figure,  will  prevent  ail  of 
^''flse  movements  except  rotation  about  the  bolt  as  au  axis.     Two 


CONSTRUCTIONS   IN   WOOD.  43 

01  more  bolts  will  prevent  this  latter,  and  consequently,  all  move- 
ment of  either  of  the  beams  upon  the  other.  A  bolt,  it  may  be 
necessary  to  say,  is  a  rod  of  iron  whose  length  is  a  little  greater 
than  the  aggregate  thickness  of  the  pieces  which  it  fastens  toge- 
ther. It  is  provided  at  one  end  with  a  solid  head,  and  at  the  other. 
with  a  few  screw  threads  on  which  turns  a  "nut,"  for  the  purpose 
of  gradually  compressing  together  the  pieces  through  which  the 
bolt  passes. 

96.  Graphical  Constructioji.  Assuming  for  simplicity's  sal^c  in 
this  and  in  most  of  these  examples,  that  the  timbers  are  a  foot  squarcj 
and  having  the  given  scale;  the  diagrams  will  generally  exi)l:iin 
themselves  sufficiently.  The  scales  are  expressed  fractionally,  adja- 
cent to  the  numbers  of  the  diagrams.  The  nut  only  is  shown  in  the 
plan  of  this  figure. 

It  is  an  error  to  suppose  that  the  nuts  and  other  small  parts  can 
be  carelessly  drawn,  as  by  hand,  without  injury  to  the  drawing, 
since  these  parts  easily  catch  the  eye,  and  if  distorted,  or  roughly 
drawn,  appear  very  badly. 

The  method  is,  therefore,  here  fully  given  for  drawing  a  nut 
accurately.  Take  any  point  in  the  centre  line,  ah,  of  the  bolt,  pro- 
duced, and  through  it  draw  any  two  lines,  cd  and  en^  at  right-angles 
to  each  other.  From  the  centre,  lay  off  each  way  on  each  line,  half 
the  length  of  each  side  of  the  nut,  say  ^  of  an  inch. 

Then,  through  the  jjoints  so  found,  draw  lines  parallel  to  the 
centre  lines  cd  and  en,  and  they  will  form  a  square  plan  of  a  nut 
1^"  on  each  side. 

In  making  this  construction,  the  distances  should  be  set  off  very 
carefully,  and  the  sides  of  the  nut  ruled,  in  very  fine  lines,  and 
exactly  through  the  points  located.  From  the  plan,  the  elevation 
is  found  as  in  PI.  II.,  Fig.  21. 

97.  Ex.  2.  A  Compound  Beam,  notched  and  "bolted.  PI. 
VII.,  Fig. 47.  Mechanical  Construction.  The  beams  represented  io 
this  figure,  are  indented  together  by  being  alternately  notched;  tha 
portions  cut  out  of  either  beam  being  a  foot  apart,  a  foot  in  length, 
and  two  inches  deep.  When  merely  laid,  one  upon  another,  they 
will  offer  resistance  only  to  being  separated  longitudinally,  and  to 
horizontal  rotation. 

The  addition  of  a  bolt  renders  the  "  compound  beam,"  thus 
formed,  capable  of  resisting  forces  tending  to  separate  it  in  all 
ways. 

Thin  pieces  are  represented,  in  this  figure,  between  the  bolt-head 
and  nut,  and  the  wood.    These  are  circular,  having  a  rounded 


44  COXSTRUCTIOXS    IN    -WOOD. 

edge, -ami  a  circular  aperture  in  the  middle  through  whicL  llie  boh 
passes.  They  are  called  "  washers,^''  and  their  use  is,  to  distribute 
the  pressure  of  the  nut  or  bolt-head  over  a  larger  surface,  so  as  not 
t(i  indent  the  wood,  and  so  as  to  prevent  a  gouging  of  the  wood 
in  tightening  the  nut,  which  gouging  would  facilitate  the  decay 
of  the  wood,  and  consequently,  the  loosening  of  the  nut. 

98.  Graphical  Construction. — The  beams  being  understood  to  be 
originally  one  foot  square,  the  compound  beam  will  be  22  inches 
deep ;  hence  draw  the  upper  and  lower  edges  22  inches  apart,  and 
from  each  of  them,  set  oiF,  on  a  vertical  line,  10  inches.  Through 
the  points,  a  and  5,  so  found,  draw  very  faint  horizontal  lines,  and 
on  either  of  them,  lay  oif  any  number  of  spaces ;  each,  one  foot  in 
length.  Through  the  points,  as  c,  thus  located,  draw  transverse 
lines  between  the  faint  lines,  and  then,  to  prevent  mistakes  in 
inking,  make  slightly  heavier  the  notched  line  which  forms  the  real 
joint  between  the  timbers. 

Tlie  use  of  the  scale  of  Jj  Cs"".tinues  till  a  new  one  is  mentioned. 

The  following  eminrical  rules  will  answer  for  determining  the 
sizes  of  nuts  and  washers  on  assumed  sketches  like  those  of  Pi.  VII., 
so  as  to  secure  a  good  appearance  to  the  diagram.  The  side  of  the 
iTut  may  be  double  the  diameter  of  the  bolt,  and  the  greater  dia- 
meter of  the  washer  may  be  equal  to  the  diagonal  of  the  nut,  plus 
twice  the  thickness  of  the  washer  itself. 

Execution. — This  is  manifest  in  this  case,  and  in  most  of  the  fol- 
lowing examples,  from  an  inspection  of  the  figures. 

99.  Ex.  3.  A  Compound  Beam,  keyed.  PI.  YIL,  Fig.  48. 
Mechanical  Construction.  The  defect  in  the  last  construction  is, 
that  the  bearing  surfaces  opposed  to  separation  in  the  direction 
of  the  length  of  the  beam,  present  only  the  ends  of  the  grain  to 
each  other.  These  surfaces  are  therefore  liable  to  be  readily 
abraded  or  made  spongy  by  the  tendency  to  an  interlacing  action 
of  the  fibres.  Hence  it  is  better  to  adopt  the  construction  given 
in  PI.  VII., Fig.  48,  where  the  "  /^eys,"  as  K,  are  supposed  to  be 
of  hard  wood,  whose  grain  runs  in  the  direction  of  the  width  of  the 
beam.  In  this  case,  the  bolts  are  passed  through  the  keys,  to  pre- 
vent them  from  slipping  out,  though  less  boring  would  be  required 
if  they  were  placed  midway  between  the  keys.  In  this  example, 
the  strength  of  the  beam  is  greatly  increased  with  but  a  very 
small  increase  of  material,  as  is  proved  in  mechanics  and  confirmed 
by  experiment. 

100.  Graphical  Construction. — This  example  difTers  from  the  last 
BO  slightly  as  to  render  a  particular  explanation  unnecessary.     The 


CONSTRUCTIONS    IN    WOOD.  45 

keys  are  12  inches  in  height,  and  6  inches  in  width,  and  are  18 
inches  apart  from  centre  to  centre.  They  are  most  accurately 
located  by  their  vertical  centre  lines,  as  AA'.  If  located  thus,  and 
from  the  horizontal  centre  line  BB',  they  can  be  completely  drawr 
before  drawing  ee'  and  rvn!.  The  latter  lines,  being  then  ])encilK'd, 
only  between  the  keys,  mistakes  in  inking  will  be  avoided. 

Execution. — The  keys  present  the  end  of  their  grain  to  view 
hence  are  inked  in  diagonal  shade  lines,  which,  in  order  to  render 
the  illuminated  edges  of  the  keys  more  distinct,  might  terminate, 
uniformly,  at  a  short  distance  from  the  upper  and  left  hand  edges. 

By  shading  only  that  portion  of  the  right  hand  edge  of  each  key, 
which  is  between  the  timbers,  it  is  shown  that  the  keys  do  no* 
project  beyond  the  front  faces  of  the  timbers. 

101.  Ex.  4.  A  Compound  Beam,  scarfed.  PI.  VII.,  Fig.  49. 
Mechanical  Construction.  This  specimen  shows  the  utie  of  a  series 
of  shallow  notches  in  giving  one  beam  a  firm  hold,  so  to  speak, 
upon  another ;  as  one  deep  notch,  having  a  bearing  surface  equal  to 
that  of  the  four  shown  in  the  figure,  would  so  far  cut  away  the 
lower  beam  as  to  render  it  nearly  useless. 

102.  G-raphical  Construction. — The  notches,  one  foot  long,  and 
tioo  inches  deep,  are  laid  down  in  a  manner  similar  to  that  described 
under  Ex.  2. 

103.  Execution. — The  keys,  since  they  present  the  end  of  the  grain 
to  view,  are  shaded  as  in  the  last  figure.  Heavy  lines  on  their  right 
hand  and  lower  edges  would  indicate  that  they  projected  beyond 
the  beam. 

Remarh. — When  the  surfaces  of  two  or  more  timbers  lie  in  the 
same  plane,  as  in  many  of  these  examples,  they  are  said  to  be 
"^ws/i''  with  each  other. 

§  III. —  Combinations  of  Timbers,  lohose  axes  make  angles  of  90° 
icith  each  other. 

104.  The  usual  w^ay  of  fastening  timbers  thus  situated,  is  by 
means  of  a  projecting  piece  on  one  of  them,  called  a  "  ?e?^o^^," 
which  is  inserted  into  a  corresponding  cavity  in  the  other,  called  a 
"  mortise.''''  The  tenon  may  have  three,  two,  or  one  of  its  sidea 
flush  with  the  sides  of  the  timber  to  which  it  belongs  ;  while  the 
mortise  may  extend  entirely,  or  only  in  part,  through  the  timbei 
in  which  it  is  made,  and  may  be  enclosed  by  that  timber  on  three 
or  on  all  sides.  [See  the  examples  which  follow,  in  w'hich  some  of 
these  case>^  arc  represented,  and  from  which  the  rest  can  be  under 

StOC'd,] 


40  CONSTRUCTIONS    IX    WOOD. 

Wlien  the  mortise  is  surrounded  on  tliree  or  0:1  two  side«,  par- 
ticularly  in  the  latter  case,  the  framed  pieces  are  said  to  b«3 
"/i«^«e(?"  together,  more  especially  in  case  they  are  of  equal  thick- 
ness, and  have  half  tlie  thickness  of  each  cut  away,  asat  PIA^II, , 
Fig.  52. 

105.  Example  1.  Two  examples  of  a  Floor  Joist  and  Sill. 
(From  a  Model.)  PL  YIl.,  Fig.  53.  Mechanical  Co7istruction. 
A — A'  is  one  sill,  B — B'  another.  CC  is  a  floor  tnnber  framed  into 
both  of  them.  At  tlie  left  hand  end,  it  is  merely  "  dropped  in,"  with 
a  tenon ;  at  the  right  hand  end,  it  is  framed  in,  with  a  tenon  and 
"  tusk,"  e.  At  the  right  end,  therefore,  it  cannot  be  lifted  out, 
but  must  be  drawn  out  of  the  mortise.  The  tusk,  e,  gives  as  great 
a  thickness  to  be  broken  off,  at  the  insertion  into  the  sill,  and  as 
much  horizontal  bearing  surface,  as  if  it  extended  to  the  full  depth 
of  the  tenon,  ^,  above  it,  while  less  of  the  sill  is  cut  away.  Thus, 
labor  and  the  strength  of  the  sill,  are  saved. 

106.  Ghxqyldcal  Construction. — \st.  Draw  ah.  2d.  On  ab  con- 
struct the  elevation  of  the  sills,  each  2^  inches  by  3  inches.  3d. 
]\Iake  the  two  fragments  of  floor  timber  with  their  upper  surfaces 
flush  with  the  tops  of  the  sills,  and  2  inches  deep.  4th.  The  mor- 
tise in  A',  is  f  of  an  inch  in  length,  by  1  inch  in  vertical  depth. 
5th.  Divide  cd  into  four  equal  parts,  of  which  the  tenon  and  tusk 
occupy  the  second  and  third.  The  tenon,  ^,  is  ^  of  an  inch  long, 
and  the  tusk,  e,  5  of  an  inch  long.     Let  the  scale  of  }  be  used. 

107.  Execution. — The  sills,  appearing  as  sections  in  elevation,  are 
shaded.  In  all  figures  like  this,  dotted  lines  of  construction  should 
be  freely  used  to  assist  in  "  reading  the  drawing,"  i.e.  in  com- 
prehending, from  the  drawing,  the  construction  of  the  thing  repre- 
sented. 

108.  Ex.  2.  Example  of  a  "Mortise  and  Tenon,"  and  of 
"Halving."  (From  a  Model.)  I'l.  VJl.,  Fig.  54.  3fecha)iicitl 
Construction.  In  this  case,  the  tenon,  AA',  extends  entirely 
through  the  piece,  CC,  into  which  it  is  fi-araed.  B  and  C  aro 
halved  together,  by  a  mortise  in  each,  A\hose  depth  equals  half 
the  thickness  of  B,  as  shown  at  B"  and  C",  and  by  the  dotted 
line,  ab. 

Graphical  Construct io7i. — Make,  l5^,  the  elevation,  A' ;  2d,  the 
plan  ;  3c?,  the  details.  B"  is  an  elevation  of  B  as  seen  when 
looking  in  the  direction,  BA.  C"  is  an  elevation  of  the  left  hand 
portion  of  CC,  showing  the  mortise  into  which  B  is  halved.  The 
dimensions  may  be  assumed,  or  found  by  a  scale,  as  noticed  below. 

109.  Execution. — The  invisible  parts  of  the  framing,  as  the  halv 


COXSTllUCTIOXS    IN    WOOD.  47 

ing,  as  seen  at  ah  in  elevation,  are  shown  in  doited  lines.  The 
brace  and  the  dotted  lines  of  construction  serve  to  show  what 
separate  figures  are  comprehended  under  tlie  general  number  (54) 
of  the  diagram.  Tlie  scale  is  }.  From  this  the  dimensions  of  the 
j)ieces  can  be  found  on  a  scale. 

110.  Ex.  3.  A  Mortise  and  Tenon  as  seen  in  tv7o  sills 
and  a  post.  Use  of  broken  planes  of  section.  (From  a 
Model.)     PI.  VII.,  Fig.  55. 

Jleckanical  Construction. — The  sills,  being  liable  to  be  drawn 
apart,  are  pinned  at  a.  The  post,  B13',  is  kept  in  its  mortise,  bb",  by 
its  own  weight ;  m  is  the  mortise  in  which  a  vertical  wall  joist 
rests.     It  is  sliown  again  in  section  near  ni'. 

111.  Graphical  Construction. — The  plan,  two  elevations,  and  a 
broken  section,  show  all  parts  fully. 

The  assemblage  is  supposed  to  bo  cut,  as  sliown  in  the  plan  by 
the  broken  line  AA'A"A"',  and  is  shown,  thus  cut,  in  the  shaded 
figure,  A'A'A"'m'.  The  scale,  which  is  the  same  as  in  Fig.  53, 
indicates  the  measurements.  At  B",  is  the  side  elevation  of  the 
model  as  seen  in  looking  in  the  direction  A' A. 

In  Fig.  55  a,  A's  obviously  equals  iVs,  as  seen  in  the  plan. 

112.  Execution. — In  the  shaded  elevation,  Fig.  55a,  the  cross-sec- 
tion, A'A'",  is  lined  as  usual.  The  longitudinal  sections  are  shaded 
by  longitudinal  shade  lines.  The  plan  of  the  broken  upper  end  of 
the  post,  B,  is  filled  with  arrow  heads,  as  a  specimen  of  a  way 
sometimes  convenient,  of  showing  an  end  view  of  a  broken  end. 

Sometimes,  though  it  renders  the  execution  more  tedious,  narrow 
blank  spaces  are  left  on  shaded  ends,  opposite  to  the  heavy  lines, 
8o  as  to  indicate  more  plainly  the  situation  of  the  illuminated  ed^-ea 
(100).  The  shading  to  the  left  of  A',  Fig.  55a,  should  be  placed 
so  as  to  distinguish  its  surface  from  that  to  the  right  of  A'. 

113.  Ex.  4.  A  Mortise  and  Tenon,  as  seen  in  timbers  so 
framed  that  the  axis  of  one  shall,  -when  produced,  be  a 
tliagonal  diameter  of  the  other.  PL  VII.,  Fig.  56.  Mechani- 
cal Construction. — In  this  case  the  end  of  the  inserted  timber  is  not 
square,  and  in  the  receiving  timber  there  is,  besides  the  moitisc,  a 
tetraedron  cut  out  of  the  body  of  that  timber. 

114.  Graphical  Construction. — D  is  the  plan,  D'  the  side  eleva- 
tion, and  D"  the  end  elevation  of  the  piece  bearing  the  tenon.  F' 
and  F  are  an  elevation  and  plan  of  the  piece  containing  the  mor- 
tise. Observe  that  the  middle  line  of  D,  and  of  D',  is  an  axis  of 
symmetry,  and  that  the  oblique  right  hand  edges  of  D  and  D'  an 
parallel  to  the  correspunding  sides  of  the  incision  in  F'. 


48 


CONSTRUCTIONS    IN    WOOD. 


§  IV. — Miscellaneous   Combinations. 

115.  Example  1.  Dowelling-.  (From  ;i  Model.)  Pl.VIT.,  Fig.57, 
Mechanical  Construction. — DoxoelUng  is  a  mode  of  fastening  by 
pins,  projecting  usually  from  an  edge  of  one  piece  into  correspond- 
ing cavities  in  another  piece,  as  seen  in  the  fastening  of  the  pails 
of  the  head  of  a  water  tight  cask.  The  mode  of  fiistening,  how- 
ever, rather  than  the  relative  position  of  the  pieces,  gives  the  name 
to  this  mode  of  union. 

The  example  shown  in  PI.  VTT. .  Fig.  57,  represents  the  braces  of 
a  roof  flaming  as  dowelled  together  Avith  oak  pins. 

1 1 6.  Graphical  Construction. — This  figure  is,  as  its  dimensions 
indicate,  drawn  from  a  model.  The  scale  is  one-third  of  an  inch 
to  a."  inch. 

t.  Draw  acJ,  witli  its  edges  making  any  angle  with  the  imagi- 
nary ground  line — not  drawn. 

2d.  At  the  middle  of  this  piece,  draw  the  pin  or  dowel.,  pp.,  ^  of 
an  inch  in  diameter,  and  projecting  f  of  an  inch  on  each  side  of  the 
piece,  ach.    This  pin  hides  another,  supposed  to  be  behind  it. 

^d.  The  pieces,  d  and  d".,  are  each  2^  inches  by  1  inch,  and  are 
shown  as  if  just  drawn  oif  from  the  dowels,  but  in  their  true  direc- 
tion, i.e.  at  right  angles  to  acb. 

Ath.  The  inner  end  of  f?is  shown  at  f?',  showing  the  two  holes, 
1^  inches  apart,  into  which  the  doicels  fit. 

Execution. — The  end  view  is  lined  as  usual,  leaving  the  dowel 
holes  blank. 

117.  Ex.  2.  A  dovetailed  Mortise  and  Tenon.  PI.  VII., 
Fig.  58.  3fechanical  Construction. — This  figure  sliows  a  species  of 
joining  called  dovetailing.  Here  the  mortise  increases  in  width  as 
it  becomes  deeper,  so  that  pieces  wliich  are  dovetailed  togetlier, 
either  at  right  angles  or  endwise,  cannot  be  pulled  directly  apart. 
The  corners  of  drawers,  for  instance,  are  usually  dovetailed ;  and 
Bometiines  even  stone  structures,  as  lighthouses,  which  are  exposed 
to  furious  storms,  have  their  parts  dovetailed  together. 

118.  Graphical  Construction. — The  skct(;hes  of  this  framing  are 
arranged  as  two  elevations.  A  bears  the  dovetail,  B  shows  the 
length  and  breadth  of  the  mortise,  and  "B"  its  depth.  A  and  B 
belong  to  the  same  elevation. 

E:cecution. — In  this  case  a  method  is  given,  of  representing  a 
hidden  cut  surface,  viz.  by  dotted  shade  lines,  as  seen  hi  the  hidden 
faces  of  the  mortise  in  B". 

119.  Leaving  now  the  examples  of  pieces  framed  together  at 
righ^  angles  let  us  consider  : — 


Pi.VII 


> 


rW^^ 


So 


SJ. 


\-^ 

•v^X/^w 

B* 

3' 

; 

1 

1 


c 


r 


OONSTRUCriOXS    IN    WO  >D.  49 

g  V,  —Pairs  of  Timbers  which  are  framed  together  obliquely  to 

each  other. 

Example  1.  A  Chord  and  Principal,     (From  a  Model.)     PI. 
V^III.,  Fig.  59.     Mechanical    Construction. — The    oblique    piece 
("  principal ")  is,  as  the  two  elevations  together  show,  of  equa 
width  with  the  horizontal  piece  ("chord,"  or  "tie  beam"),  and  i 
fi  am(>d  into  it  so  as  to  prevent  sliding  sidewise  or  lengthwise. 

Neither  can  it  be  lifted  out,  on  account  of  the  bolt  which  is  made 
to  pass  perpendicularly  to  the  joint,  ac,  and  is  "chipped  up"  atj!?p, 
so  as  to  give  a  ilat  bearing,  parallel  to  ac,  for  the  nut  and  bolt-head. 

120.  Graphical  Construction. — \st.  Draw^jfZe/  2d.  Lay  oflf  cfe 
^13  inches;  Sd.  Make  e'ea  —  ^0'^\  4th.  At  any  point,  e',  draw  a 
perj)endicular  to  ee\  and  lay  off  upon  it  9  inches — the  perpendicular 
width  of  e'ea/  5th.  Makeec=:4  inches  and  perpendicular  to  e'e/ 
bisect  it  and  complete  the  outlines  of  the  tenons,  and  the  shoulder 
anc';  Qth.  To  draw  the  nut  accurately,  joroceed  as  in  PI.  VII.,  Fig. 
4<;-47,  placing  the  centre  of  the  auxiliary  projection  of  the  nut  in 
the  axis  of  the  bolt  produced,  &c.  (40)  (96).  b  represents  the  bolt 
hole,  the  bolt  being  shown  only  on  one  elevation. 

121.  Ex.  2.  A  Brace,  as  seen  in  the  angle  bet-ween  a 
"post"  and  "girth."  (From  a  Model.)  PL  VIIL,  Fig.  60. 
Jfechanical  Construction. — PP' is  the  post,  GG' is  tKe  girth,  and 
Pi'B"  is  the  brace,  having  a  truncated  tenon  at  each  end,  Avhich  rests 
in  a  mortise.  When  the  brace  is  quite  small,  it  has  a  shoulder  on 
one  side  only  of  the  tenon,  as  if  B'B"  were  sawed  lengthwise  on 
a  line,  oo'. 

122.  Graphical  Construction. — To  show  a  tenon  of  the  brace 
clearly,  the  girth  and  brace  together  are  represented  as  being 
drawn  out  of  the  post.  1st.  Draw  the  post.  2d.  Half  an  inch 
below  the  top  of  the  post,  draw  the  girth  2^  inches  deep.  3c?.  Frora 
a,  lay  off  ab  and  a  each  4  inches,  and  draw  the  brace  1  inch  wide„ 
UJi.  Make  cd  equal  to  the  adjacent  mortise;  viz.  1^  inches;  make 
<?(?=1  inch,  and  erect  the  perpendicular  at  e  till  it  meets  be,  &c. 
Tlie  dotted  projecting  lines  show  the  construction  of  B"  and  of  the 
plan.  At  e"  is  the  vertical  end  of  the  tenon  e.  On  each  side  of 
c",  are  the  vertical  surfaces,  show^n  also  at  cd.  Let  B'  also  be  pro- 
jected on  a  plane  parallel  to  be. 

123.  Ex.  3.  A  Brace,  -with  shoulders  mortised  into  the 
post.  PI.  Vlll.,  Fig.  61.  This  is  the  strongest  way  of  framing  a 
brace.  For  the  rest,  the  figure  explains  itself  Observe,  however, 
that  while  in  Fig.  59,  the  head  of  the  tenon  and  shoulder  is  perpendi* 


50  CONSTRUCTTONS   IN   WOOI). 

cular  to  the  oblique  piece,  here,  where  that  piece  is  framed  nito  a 
vertical  post,  its  head  nu  is  perpendicular  to  the  axis  of  the  post. 
In  Fig.  61,  moreover,  the  auxiliary  plane  on  which  the  brace 
alone  is  projected,  is  parallel  to  the  length  of  the  brace,  as  is 
shown  l)y  the  situation  of  B,  the  auxiliary  projection  of  the  brace, 
and  by  the  direction  of  the  projecting  lines,  as  n-'.  P'  is  the 
elevation  of  P,  as  seen  in  the  direction  nn\  and  with  the  brace 
removed. 

124.  Ex.  4.  A  "Shoar."  PI.  VIII.,  Fig.  G2.  Mechanical  Coiv 
8tnictio7i. — ^A  "  shoar  "  is  a  large  timber  used  to  prop  up  earth  or 
buildings,  by  being  framed  obliquely  into  a  horizontal  beam 
and  a  stout  vertical  post.  It  is  usually  of  temporary  use,  during 
the  construction  of  permanent  works ;  and  as  respects  its  action, 
it  resists  compression  in  the  direction  of  its  length.  To  give  a 
large  bearing  surface  without  cutting  too  far  into  the  vertical  tim- 
ber, it  often  has  two  shoulders.  The  surface  at  ab  is  made  verti- 
cal, for  then  the  fibres  of  the  post  are  unbroken  except  at  cJ,  while 
if  the  upper  shoulder  were  shaped  as  at  aclb,  the  fibres  of  the  trian- 
gular portion  dbc  would  be  short,  and  less  able  to  resist  a  longitu- 
dinal force. 

125.  The  Graphical  construction  is  evident  from  the  figure, 
which  is  in  two  elevations,  the  left  hand  one  showing  the  post 
only. 

Execution. — The  vertical  surface  at  ab  may,  in  the  left  hand 
elevation,  be  left  blank,  or  shaded  with  vertical  lines  as  in  PL  VII., 
Fig.  55  a. 

§  VI. —  Combinations  of  Timbers  whose  axes  m,a7ce  angles  q/*180** 
xoith  each  other. 

126.  Timbers  thus  framed  are,  in  general,  said  to  be  spliced.  Six 
forms  of  splicing  are  shown  in  the  following  figures. 

Example  1.  A  Halved  Splicing,  pinned.  PI.  VIIL,Fig.  63. 
The  mechanical  construction  is  evident  from  the  figure.  When 
boards  are  lapped  on  their  edges  in  this  way,  as  in  figure  69,  they 
are  said  to  be  "  rabbetted.'''' 

1 27.  Graphical  Construction. — After  drawing  the  lines,  12  inchea 
apart,  which  represent  the  edges  of  the  timber,  drop  a  perpendicu- 
lar of  6  inches  in  length  from  any  point  as  a.  From  its  lower  ex- 
tremity, draw  a  horizontal  line  12  inches  in  length,  and  from  c,  drop 
the  ])erpendicular  cb^  which  comj)letes  the  elevation.  In  the  plan, 
the  joint  at  a  will  be  seen  as  a  full  line  a'a",  and  that  at  5,  being 
hidden  is  represented  as  a  dotted  line,  at  h' 


i 


t;ONSTRUCTI0NS    IN   WOOD.  51 

Draw  a  dingonal,  ah" ^  divide  it  into  four  equal  pai ts,  and  taka 
the  first  and  third  j)oints  of  division  for  the  centres  of  two  pins, 
having  each  a  radius  of  three-fourths  of  an  inch. 

Execution. — The  position  of  the  heavy  lines  on  these  figures  in 
too  obvious  to  need  remark. 

128.  Ex.  2.  Ton^ing  and  Grooving;  and  Mortise  and 
Tenon  Splicing.  I'l.  VllL,  Fig.  G4.  Boards  united  at  then- 
edges  in  this  way,  as  shoAvn  in  PI.  VIII.,  Fig.  70,  are  said  to  be 
tongued  and  grooved. 

Drawing,  as  before,  the  plan  and  elevation  of  a  oeam,  a  foot 
square,  divide  its  depth  an,  at  any  point  a,  into  five  equal  parts. 
Take  the  second  and  fourth  of  these  parts  as  the  width  of  the 
tenons,  which  are  each  a  foot  long. 

The  joint  at  a  is  visible  in  the  plan,  the  one  at  h  is  not.  Let  a'd 
be  a  diagonal  line  of  the  square  a'd.  Divide  a'd  into  three  equal 
parts,  and  take  the  points  of  division  as  centres  of  inch  bolts,  with 
heads  and  nuts  2  inches  square,  and  washers  of  If  inches  radius.  To 
place  the  nut  in  any  position  on  its  axis,  draw  any  two  lines  at  right 
angles  to  each  other,  through  each  of  the  bolt  centres,  and  on  each, 
lay  off  1  inch  from  those  centres,  and  describe  the  nut.  Project  up 
those  angles  of  the  nut  which  are  seen ;  viz.  the  foremost  ones, 
make  it  1  inch  thick  in  elevation,  and  its  washer  \  an  inch  .thick. 

In  this,  and  all  similar  cases,  the  head  of  the  bolt,  5,  would  not 
have  its  longer  edges  necessarily  parallel  to  those  of  the  nut.  To 
give  the  bolt  head  any  position  on  its  axis,  describe  it  in  an  auxiliary 
plan  just  below  it. 

129.  Ex.  3.  A  Scarfed  Splicing,  strapped.  PI.  VIII.,  Fig.  65. 
Mechanical  Construction. — While  timbers,  framed  together  as  in 
the  two  preceding  examples,  can  be  directly  slid  apart  when  their 
connecting  bolts  are  removed;  the  timbers,  fi-amed  as  in  the  present 
example,  cannot  be  thus  separated  longitudinally,  on  account  of  the 
dovetailed  form  of  the  splice.  Strapping  makes  a  firm  connexion, 
but  consumes  a  great  deal  of  the  uniting  material. 

130.  Graphical  Construction. — After  drawing  the  outlines  of  the 
side  elevation,  make  the  perpendiculars  at  a  and  a',  each  8  inches 
long,  and  make  them  18  inches  apart.  One  inch  from  a,  make  the 
strap,  5s',  2^  inches  wide,  and  projecting  half  an  inch — i.  e.  its 
thickness — over  the  edges  of  the  timber.  The  ear  through  which 
the  bolt  passes  to  bind  the  strap  round  the  timber,  projects  two 
inches  above  the  strap.  Take  the  centre  of  the  ear  as  the  centre 
of  the  bolt,  and  on  this  centre  describe  the  bolt  head  1^  inches 
square. 


52  CCXSTUUCTIONS    IN    WOOD. 

In  the  i)]an,  the  eai's  of  the  strap  are  at  an}'  indefinite  distanci 
apart,  depending  on  the  tightness  of  the  nut,  s. 

A  fragment  of  a  similar  strap  at  the  other  end  of  the  scarf,  is 
shown,  with  its  risible  ends  on  the  bottom,  near  a',  and  back,  at  r 
of  the  beam. 

J^xecutlon. — The  scarf  is  dotted  wliere  it  disappears  behind  tho 
strap  ;  and  so  are  the  hidden  joint  at  a',  and  the  fragment  of  tho 
Becond  strap,  as  shown  in  plan.  These  examples  may  advan- 
tageously be  drawn  by  the  student  on  a  scale  of  ^V-  Care  must 
then  be  exercised  in  making  the  large  broken  ends  neatly,  in  large 
splinters,  edged  with  fine  ones. 

131.  Ex.4.  A  Scarfed  Splice,  bolted.  PI.  MIL,  Fig.  60. 
Mechanical  Construction. — AA'  is  one  timber.  BB'  is  the  other. 
Each  is  cut  oti'  as  at  aea",  forming  a  pointed  end  which  prevents 
latei-al  displacement,  i/and^^^are  the  ends  of  transverse  keys, 
which  afford  good  bearing  surfaces.  See  the  same  on  PI.  XIII., 
Fig.  105,  which  shows  the  arrangement  plainly. 

132.  Graphical  Constrxction. — Let  each  timber  be  1  foot  square 
and  let  the  scale  be  1  foot  to  1  inch.  Draw  the  outlines  accord- 
ingly, and,  assuming  a',  make  a'5'  =  3  feet;  drop  the  perpendicular 
b'c\  and  draw  a  line  of  construction  a'c'.  From  a'  lay  oflf  4  inches 
on  a'c'  and  divide  the  rest  of  a'c'  into  three  equal  parts,  to  get  the 
size  of  the  equal  spaces  c'k,fd  and  gv\  and  at  c'  and  k  draw  per- 
pendiculars to  a'c\  above  it  and  2  inches  long.  Divide  a'k  into 
two  equal  parts  at  J,  and  from  k  and  d  lay  off  2  inclies  on  a'c' 
towards  a'  and  complete  the  keys,  as  shown,  kf  ahove  a'c',  and  dg 
below  it;  also  the  joints  v'g^  dk,  etc.,  of  the  splice,  Avhere  a'v'  is 
perpendicular  to  a'c'  and  2  inclies  long. 

Now,  in  plan,  pi-oject  down  a'  at  a  and  a"  and  draw  ae  and  a"e 
at  60°  with  a"c,  and  do  the  same  with  c',  as  shown.  Project  up  e 
to  c'  and  draw  e's'  parallel  to  a'v'  till  it  meets  c'g  produced.  Then 
proji'cl  down  v'  at  v  and  v",  and  s'  at  ft,  and  draw  sv  and  sv", 
which,  of  course,  will  not  be  paralh'l  to  ae  and  a"e,  since  se — s'e'  is 
shorter  than  av — a'v'.  Thus  au  se  a"v" — a'v'  s'e'  is  the  obliquely 
pointed  end  of  the  piece  AA'.     BB'  is  similarly  pointed,  as  shown. 

Add  the  bolts,  nuts  and  washers,  nw' and  mni'^in  any  convenient 
position,  whicli  will  complete  the  construction. 

liJxecution. — The  keys  are  shaded.  The  hidden  cut  surfaces  of 
the  notches  aie  shaded  in  dotted  shade  lines,  and  the  hidden  joints 
are  dotted. 

133.  p]x.  5.  A  Compound  Beam,  Avith  one  of  the  compo 
Dent  beams  "fished."  PI.  \'I11.,  l•'i'^  67.  Mechanical  Construo 


J 


CONSTRUCTIONS    IN   WOOD.  53 

tian. — ^The  mode  of  union  called  fishing^  consists  in  uniting  two 
pieces,  end  to  end,  by  laying  a  notched  piece  over  the  joint  and 
bolting  it  through  the  longer  pieces. 

The  figure  shows  this  mode  as  applied  to  a  compound  beam, 
i.  e.  to  a  beam  "  huilt "  of  several  pieces  bolted  and  keyed  toge- 
ther. The  order  of  construction  is  as  follows — taking  for  a  scale 
f  of  an  inch  to  a  foot. 

134.  Graphical  Construction. — \st.  The  outside  lines  of  the  plan 
are  two  feet  apart,  the  outside  pieces  are  each  4^  inches  wide, 
and  the  interior  ones  5^  inches  ;  leaving  four  inches  for  the  sum 
of  the  three  equal  spaces  between  the  four  beams. 

2c?.  Let  there  be  a  joint  at  a'.  Lay  off  3  feet,  each  side  of  a'  foi 
the  length  of  the  "/sA." 

dd.  The  straight  side  of  this  piece  is  let  into  the  whole  piece,  5, 
two  thirds  of  an  inch,  and  into  a',  1  inch,  making  its  thickness  3 
inches. 

Ath.  At  a\  it  is  two  inches  thick,  i.  e.  at  a\  the  timber,  a',  is  of 
its  full  thickness.  The  fish,  c,  is  2  inches  thick  for  the  space  of  one 
foot  at  each  side  of  a'.  The  notches  at  d  and  e  are  each  1  inch 
deep,  dd'  and  ee'  each  are  one  foot,  and  the  notches  at  d'  asd  e 
are  each  1  inch  deep.  The  remaining  portions  of  the  fish  a^'e  1 
foot  long,  and  3  inches  wide. 

hth.  Opposite  tQ  these  extreme  portions,  are  keys,  1  foot  by  3 
inches,  in  the  spaces  between  the  other  timbers,  and  setting  an  equal 
depth  into  each  timber. 

Gth.  1\\  elevation,  only  the  timber  a'  is  seen — 1  foot  deep.  Four 
bolts  pass  through  the  keys.  h'b"=^b  feet,  and  b'  is  three  inches  from 
the  top,  and  from  kk\  the  left  hand  end  of  the  fish,  n'n" =5  feet, 
and  n'  is  3  inches  above  the  bottom  of  the  timber,  and  9  inches  from 
kh' .  The  circular  bolt  head  is  one  inch  in  diameter,  and  its  washer 
3^  inches  diameter  and  h,  an  inch  thick.  The  thickness  of  the  bolt 
head,  as  seen  in  plan,  is  f  inch.  The  nuts,  mi,  are  1^  inches  square, 
and  ^  an  inch  thick,  and  the  bolts  are  half  an  inch  in  diameter. 

136.  Tne  several  nuts  would  naturally  be  found  in  various  posi- 
tions on  their  axes.  To  construct  them  thus  with  accuracy,  as  seeu 
in  the  plan,  one  auxiliary  elevation,  as  N,  is  sufficient.  N,  and  its 
centre,  may  be  projected  upon  as  many  planes — xy — as  there  are 
different  positions  to  be  represented  in  the  plan,  each  plane  being 
supposed  to  be  situated,  in  reference  to  N,  as  some  nut  in  the  plan 
18,  to  its  elevation.  Then  transfer  the  points  on  a-y,  &c.,  to  the 
outside  of  the  several  w?*^washers,  placing  the  projection  of  the 
centro  lines  of  the  bolts  in  the  plan,  a«k  lines  of  reference. 


54  CONSTRUCTIONS    IN   WOOD. 

JiJxecution. — Th(j  figure  explains  itself  in  tliis  respect. 

l:!G.  Ex.  6.  A  vertical  SpUce.  PI.  VIII.,  Fig.  68.  Mecha, 
nical  Construction. — This  splice  is  formed  of  two  prongs  at  oppo- 
site corners  of  each  piece,  embraced  by  corresponding  notches  in 
the  other  piece.  Thus  in  the  piece  B',  the  visible  prong,  as  seen  in 
elevation,  is  a  truncated  triangular  pyramid  whose  horizontal  base 
18  abc — a'c\  and  whose  oblique  base  is  enc — e'n''c'.  Besides  the 
four  prongs,  two  on  each  timber,  there  is  a  flat  surface  ahfq — c'a'q\ 
well  adapted  to  receive  a  vertical  pressure,  since  it  is  equal  upon, 
and  common  to,  both  timbers. 

137.  G-rajyJiical  Construction. — To  aid  in  understanding  this 
combination,  an  oblique  projection  is  given  on  a  diagonal  plane, 
parallel  to  PQ. 

1st.  Make  the  plan,  acfq,  mth  the  angles  of  the  interior  square 
in  the  middle  of  the  sides  of  the  outer  one.  2c?.  Make  the  distances, 
as  ce=:2  inches  and  draw  ew,  &c.  Zd.  Make  c'n  z^c'n"  15  inches, 
and  draw  short  horizontal  lines,  n'e',  &c.,  on  which  project  e,  &c., 
after  which  the  rest  is  readily  completed. 

138.  Execution. — Observe,  in  the  plans,  to  change  the  direction 
of  the  shade  lines  at  every  change  in  the  position  of  the  surfac*  of 
the  wood. 


CHAPTER  m. 

CONSTRUCTIONS   IN   METiL. 

139.  KvaTsiTi.1,  \.   An  end  view  of  a    Railroad  Rail.     PI. 

\'III.,  Fig,  Ti.  ChuprAcal  Construction. — 1st.  Draw  a  vertical  centre 
line  AA',  and  make  AA'=3f  inches.  2d.  Make  A'b=A'b'  =  2 
iiiclies.  3d.  Make  Ac = Ac' =  1  inch.  4th.  Describe  two  quadrants, 
of  which  c'd  is  one,  with  a  radius  of  half  an  inch.  5th.  With  /), 
half  an  inch  from  AA',  as  a  centre,  and  pd  as  a  radius,  describe  an 
arc,  dr,  till  it  meets  a  vertical  line  through  e.  Qth.  Draw  the  tangent 
rs.  1th.  Draw  a  vertical  line,  as  pt'.,  \  an  inch  from  Kh! .,  on  each 
side  of  AA'.  8^A.  Bisect  the  angle  rst'  and  note  s',  where  the 
bisecting  line  meets  the  radius,  />r,  produced.  9^7i.  "With  s'  as  a 
centre,  draw  the  arc  rt'.  \Qth.  At  h  and  h'  erect  perpendiculars, 
each  one  fourth  of  an  inch  high.  Wth.  Draw  quadrants,  as  q't.^ 
tangent  to  these  perpendiculars  and  of  one  fourth  of  an  inch  radius. 
\2th.  Draw  the  horizontal  line  ^y.  \Mh.  Make  ni'=nu  and  describe 
the  arc  t'v.  lith.  Repeat  these  operations  on  the  other  side  of  the 
centre  line,  AA'. 

Mcecutiofi. — Let  the  construction  be  fully  shown  on  one  side  of 
the  centre  line. 

140.  Ex.2.  An  end  elevation  ofa  Compound  Rail.  PI.  VIIL, 
Fig.  72.  '3fechanical  Construction. — The  compound  rail,  is  a  rail 
formed  in  two  parts,  which  are  placed  side  by  side  so  as  to  break 
joints,  and  then  riveted  together.  As  one  half  of  the  rail  is  Avhole, 
at  the  points  where  a  joint  occurs  on  the  other  half,  the  noise  and 
jar,  observable  in  riding  on  tracks  built  in  the  ordinaiy  manner,  are 
both  obviated  ;  also  "  chairs,"  the  metal  supports  which  receive  the 
ends  of  the  ordinary  rails,  may  be  dispensed  with,  in  case  of  the 
use  of  the  compound  rail. 

In  laying  a  compound  rail  on  a  curve,  the  holes,  through  which 
the  bolts  pass,  may  be  drawn  past  one  another  by  the  bending 
of  the  rails.  To  allow  for  this,  these  holes  are  "  slotted,''  as  it  is 
termed,  i.  e.  made  longer  in  the  direction  of  the  length  of  the  rail. 

141.  Graphical  Construction. — \st.  Make  <y=4  inches,  'id. 
Bisect  ty  at  w,  and  erect  a  perpendicular,  wa,  of  3^  inclies.     2,d, 


56  COXSTRUCTIONS    IN    METAL. 

Mane,  snocessi\ely,  tfr=^  an  inch ;  from  r  to  cb  =  2  iuches;  from  ?< 
to  nh  —  l  of  an  inch;  and  to  ge  =  2^  inches.  4(h.  For  tlie  several 
W'idtlis  of  the  interior  parts,  make  bc—^  of  an  inch,  and  «7  and  e 
each  f  of  an  inch,  from  ua ;  nh—ge^  and  ro=%  of  an  inch.  5th. 
To  locate  tlie  outlines  of  the  rail,  make  ms^  the  Hat  top,  called  the 
tread  of  the  rail, =2  inches,  half  an  inch  below  this,  make  the  width, 
/c?,  3  inches ;  and  make  the  part  through  which  the  rivet  passes,  1 1 
inches  thick,  and  rounded  into  the  lower  flange  which  is  f  of  an 
inch  thick. 

The  rivet  has  its  axis  If  inches  from  ty.  Its  original  head,  5,  ia 
conical,  with  bases  of — say  1  inch,  and  f  of  an  inch,  diameter;  and 
is  half  an  inch  thick.  The  other  head, />,  is  made  at  pleasure,  being 
roughly  hammered  down  while  the  rivet  is  hot,  during  the  process 
of  track-laying.     A  thin  washer  is  shown  under  this  head. 

142.  Ex.  3.  A  "  Cage  Valve,"  from  a  Locomotive  Pump. 
ri.  A'lll.,  Fig.  73-74. — Mechanical  Construction. — Tliis  valve  is 
made  in  three  pieces,  viz.  the  valve  proper,  Fig.  74  ;  the  cage  con- 
taining it.  Abb';  and  the  flange  bb'c  ;  whose  cylindrical  aperture — 
shown  in  dotted  lines — being  smaller  than  the  valve,  confines  it. 
The  valve  is  a  cup,  solid  at  the  bottom,  and  makes  a  water  tight 
joint  with  the  upper  surface  of  the  flange,  inside  of  the  cage.  The 
whole  is  inclosed  in  a  chamber  communicating  with  the  pump 
barrel,  and  with  the  tender,  or  the  boiler,  according  as  we  suppose 
it  to  be  the  inlet  or  outlet  valve  of  the  pump.  This  chamber  nukes 
a  water  tight  joint  with  the  circumference  of  the  flange  bb.' 

Suppose  the  valve  to  be  the  latter  of  the  two  just  named.  The 
"  plunger"  of  the  pump  being  forced  in,  the  water  shuts  the  inlet 
valve,  and  raises  the  outlet  valve,  and  escapes  between  the  bars  of 
the  cage  into  the  chamber,  and  from  that,  by  a  pipe,  into  fhe  boiler, 

143.  Graphical  Construction. — Scale  full  size.  Make  the  plan 
Brst,  where  the  six  bars  are  equal  and  equidistant, with  radial  sides. 
Project  them  into  the  elevation;  as  in  Prob.  14,  Div.  1.;  taking 
care  to  note  whether  any  of  the  bars,  as  E,  on  the  back  part  of  the 
cage  can  be  seen  above  the  valve,  CD,  and  between  the  front  bars, 
as  F  and  G.  The  diameters  of  the  circles  seen  in  the  cage  are,  in 
order,  from  the  centre,  1^,  2J,  2||-,  and  4  inches.  The  thickness  of 
the  valve.  Fig.  74,  is  -j\  of  an  inch,  its  outside  height  If  inches, 
and  the  outside  diameter  2^  inches.  The  diameter  of  the  aperture 
in  the  flange  is  1|  inches,  its  length  |-  of  an  inch,  and  the  height  of 
the  whole  cage  is  3-^^  inches. 

144.  Execution. — Observe  carefully  the  position  of  the  heavy 
lines.     The  section,  Fig.  74,  being  of  metal,  is  finely  shaded. 


J 


CONSTRUCTIONS    IN    METAL.  57 

745.  Ex.  4.    An  oblique  elevation  of  a  Bolt  Head.     PL 

VIII.,  Fig. 75.  Let  PQ  be  the  intersection  of  two  vertical  planes, 
at  right  angles  with  each  other;  and  let  RS  be  the  intersection, 
with  the  vertical  plane  of  the  paper  to  the  left  of  PQ,  of  a  ])lane 
wliich,  in  space,  is  parallel  to  the  square  top  of  the  bolt  head.  On 
such  a  plane,  a  plan  view  of  the  bolt  head  may  be  made,  showing 
two  of  its  dimensions  in  their  real  size;  and  on  the  plane  above  RS, 
the  thickness  of  the  bolt  head,  and  diameter  of  the  bolt,  are  shown 
in  their  real  size. 

Below  RS,  construct  the  plan  of  the  bolt  head,  with  its  sides 
making  any  angle  with  the  ground  line  RS.  Project  its  corners  in 
perpendiculars  to  RS,  giving  the  left  hand  elevation,  whose  thick- 
ness is  assumed. 

146.  The  fact  that  the  projecting  lines  of  a  point,  form,  in  the 
drawings,  a  perpendicular  to  the  ground  line,  is  but  a  special  case 
of  a  more  general  truth,  which  may  be  thus  stated. — When  an 
object  in  space  is  projected  upon  any  two  planes  which  are  at  right 
angles  to  each  other,  the  projecting  lines  of  any  point  of  that  object 
form  a  line,  in  the  drawing,  perpendicular  to  the  intersection  of  the 
two  planes. 

147.  To  apply  the  foregoing  principle  to  the  present  problem; 
it  appears  that  each  point,  as  a",  of  the  right  hand  elevation,  will 
be  in  a  line,  a'a!\  perpendicular  to  PQ,  the  intersection  of  the  two 
vertical  planes  of  projection. 

Remembering  that  PQ  is  the  intersection  of  a  vertical  plane — 
perpendicular  to  the  plane  of  the  paper — with  the  vertical  plane 
of  the  paper,  and  observing  that  the  figure  represents  this  plane  aa 
being  revolved  around  PQ  towards  the  left,  and  into  the  plane  of 
the  paper,  and  observing  the  arrow,  which  indicates  the  direction 
in  which  the  bolt  head  is  viewed,  it  appears  that  the  revolved  ver- 
tical plane,  has  been  transferred  from  a  position  at  the  left  of  the 
plan  acne^  to  the  position,  PQ,  and  that  the  centre  line  ^?f",  must 
appear  as  far  from  PQ  as  it  is  in  front  of  the  plane  of  the  paper — 
i.  e.  e'V=ew,  showing  also,  that  ase — e'  is  in  the  plane  of  the  paper, 
its  projection  at  e"  must  be  in  PQ,  the  intersection  of  the  two  ver- 
tioal  planes. 

Similarly,  the  other  corners  of  the  nut,  as  c",  w",  &c.,  are  laid  off 
either  from  the  centre  line  txi'\  or  from  PQ.  Thus  v"c''=vc,  oi 
v"'c"  =  sc.     The  diameter  of  the  bolt  is  equal  in  both  elevations. 

148.  Other  supposed  positions  of  the  auxiliary  plane  PQ  may  be 
assumed  by  the  student,  and  the  corresponding  construction  worked 
out.     Thus,  the  primitive  position  of  PQ  maybe  at  the  right  of  the 


58  CONSTRUCTIOXS    IX    METAU 

Dolt  head,  and  that  may  be  viewed  in  the  opposite  direction  frona 
ihal  indicated  by  tlie  arrow. 

149.  Ex.  5.  A  "  Step"  for  the  support  of  an  oblique  tim 
bex.  Pi.  V'lll.,  Fig.  7G.  MecJianical  Construction. — It  will  be 
trec^iiently  obsei'ved,  in  the  framings  of  bridges,  that  there  are 
certam  timbers  whose  edges  have  an  oblique  direction  in  a  vertical 
plane,  while  at  their  ends  tliey  abut  against  horizontal  timbers,  not 
directl)^,  for  that  would  cause  them  to  be  cut  off  obliquely,  but 
through  the  medium  of  a  prismatic  block  of  wood  or  iron,  so 
shaped  that  one  of  its  faces,  as  ab — n'b\  Fig.  76,  rests  on  the  hori- 
zontal timber,  while  another,  as  ac — e"d''c",  is  perpendicular  to  the 
oblique  timber. 

To  secure  lightness  with  strength,  the  step  is  hollow  underneath, 
and  strengthened  by  ribs,  r)\  The  holes,  h'h",  allow  the  passage 
of  iron  rods,  used  in  binding  together  the  partsof  the  bridge.  These 
holes  are  here  prolonged,  as  at  h,  forming  tubes,  which  extend  partly 
or  wholly  through  the  horizontal  timber  on  which  the  step  rests,  in 
order  to  hold  the  step  steadily  in  its  place. 

150.  Hemarh.  When  the  oblique  timber,  as  T,  Fig.  76  (a),  seta 
into^  rather  than  upon,  its  iron  support  S,  so  that  the  dotted  lines, 
ab  and  ci,  represent  the  ends  of  the  timber,  the  support,  S,  is  called 
a  shoe. 

151.  Graphical  Construction. — In  the  plate,  ahc  is  the  elevation, 
and  according  to  the  usual  arrangement  would  be  placed  above  the 
plan,  e"n"c\  of  the  top  of  the  step,  a'b'e  is  the  plan  of  the  under 
Bide  of  the  step,  showing  the  ribs,  Sec.  A  line  through  nm  is  a 
centre  line  for  this  plan  and  for  the  elevation.  A  line  througli  the 
middle  point,  r,  of  m?i,  is  another  centre  line  for  the  plan  of  the 
bottom  of  the  step.  Having  chosen  a  scale,  the  position  of  the 
centre  lines,  and  the  arrangement  of  the  figures,  the  details  of  the 
construction  may  be  left  to  the  student. 

152.  Ex,  6.  A  metallic  steam  tight  "Packing,"  for  the 
"stuffing  boxes"  of  piston  rods.  Tl.  Mil.,  Jig.  77. 

Meclianical  Construction. — Attached  to  that  end,  J,  of  Fig.  77  (a), 
of  a  steam  cylinder,  for  instance,  at  which  the  piston  rod,  p,  entera 
It,  is  a  cylindrical  projection  or  "  neck,"  n,  having  at  its  outer  end 
a  flange,  f/\  through  which  two  oi-  more  bolts  pass.  At  its  inner 
end,  at^:),  this  neck  fits  the  piston  rod  quite  close  for  a  short  space. 
Tlie  internal  diameter  of  the  remaining  portion  of  the  neck  is  suffi- 
cient to  receive  a  ring,  rr,  which  fits  the  piston  rod,  and  has  on  its 
outer  edge  a  flange,  <,  by  which  it  is  fastened  to  the  flange,  Jf,  on 
the  neck  of  the  cylinder  by  screw  bolts.     The  remaining  hcillow 


CONSTRUCTIONS   IN   METAL.  59 

space,  s,  bet\veen  the  ring  or  "  gland,"  ^r,  and  the  inner  end  of  tlie 
neck,  is  usually  filled  with  some  elastic  substance,  as  picked  hemp, 
which,  as  held  in  place  by  the  gland,  tr,  makes  a  steam  tight  joint; 
which,  altogether,  is  called  the  "stuffing  box." 

153.  The  objection  to  this  kind  of  packing  is,  that  it  requires  so 
frequent  renewals,  that  much  time  is  consumed,  for  instance  in  raii 
road  repair  shops,  in  the  preparation  and  adjustment  of  the  packing. 
To  obviate  this  loss  of  time,  and  perhaps  because  it  seems  mon 
neat  and  trim  to  have  all  parts  of  an  engine  metallic,  this  metallic 
packing.  Fig.  V7,  was  mvented.  ABC — A'B'  is  a  cast  iron  ring, 
cylindrical  on  th(?  outside,  and  having  inlaid,  in  its  circumference, 
bands,  tt\  of  soft  metal,  so  that  it  may  be  squeezed  perfectly  tight 
into  the  neck  of  the  cylinder.  The  inner  surface  of  this  ring  is  coni- 
cal, and  contains  the  packing  of  block  tin.  This  packing,  as  a  whole, 
is  also  a  ring,  whose  exterior  is  conical,  and  fits  the  inner  side  of  the 
iron  ring,  and  whose  interior, yA;c,  is  cylindrical,  fitting  the  piston 
rod  closely. 

154.  For  adjustment,  this  tin  packing  is  cut  horizontally  into 
three  rings,  and  each  partial  ring  is  then  cut  vertically,  as  shown  in 
the  figure,  into  two  equal  segments,  ahcd — a'b'c'd',  is  one  segment 
of  tl  e  upjDer  ring;  efgh — e'f'f"g'h\  is  one  segment  of  the  middle  ring, 
ind  rjM — r'fk'n'l,  is  one  segment  of  the  lower  ring.  The  segments 
of  each  ring,  it  therefore  appears,  break  joints  with  each  of  the  other 
rings.  Three  of  the  segments,  one  in  each  partial  ring,  are  loose, 
while  the  other  three  are  dowelled  by  small  iron  pins,  parallel  to 
the  axis  of  the  whole  packing. 

155.  Operation. — Suppose  the  interior,  ^/tc,  of  the  packing  to  bo 
of  less  diameter  than  the  piston  rod,  which  it  is  to  surround.  By 
drawing  it  partly  out  of  its  conical  iron  case,  the  segments  forming 
each  ring  can  be  slightly  separated,  making  spaces  at  a5,  &c.,  which 
will  increase  the  internal  diameter,  so  as  to  receive  the  piston  rod. 
When  in  this  position,  let  the  gland,  ti\  be  brought  to  bear  on  the 
packing,  and  it  will  be  firmly  held  in  place ;  then,  as  the  packing 
gradually  wears  away,  the  gland,  by  being  pressed  further  into  the 
neck,  will  press  the  packing  further  into  its  conical  seat,  which 
Avill  close  up  the  segments  round  the  piston  rod. 

156.  Graphical  Construction. — Let  the  scale  be  from  one  half 
to  the  whole  original  size  of  the  packing  for  a  locomotive  valve 
chest.  C  is  the  centre  for  the  various  circles  of  the  plan,  and  DD'. 
projected  up  from  C,  is  a  centre  line  for  portions  of  the  elevation 
C/=-j2g-  of  an  inch;  Ce=::l-^  inches;  CA=1^  inches.  A'n=2 
inches,  and  K's'—-^  of  an  inch.     The  iron  case  being  constructec' 


60 


COXSTKUCTIONS    IN    ^^ETA1. 


from  tliese  inoasurements,  the  rings  must  be  located  so  tliaty*/)',  foj 
instance,  shall  be  =  j\  of  an  inch  ;  and  then  let  tlie  thickness,  /'/"', 
of  cacli  segment  be  \}  of  an  inch.  Tliese  dimensions  and  the  con- 
sequent ai-i-angenients  of  the  rings  \v ill  give  spaces  between  the  seg. 
ments,  as  at  ab,  of  i  of  an  inch  ;  though  in  fact,  as  this  space  is 
variable,  there  is  no  necessity  for  a  precise  measurement  for  it.  In 
the  plan,  there  are  shown  one  segment,  and  a  fragment,  abef^  of 
another,  in  the  u]iper  ring;  one  segment  and  two  fragments  of  tht 
middle  ring,  and  both  segments  of  the  lower  ring,  with  the  whoh^ 
of  the  iron  case.  The  elevation  shows  one  of  the  halves  of  eacb 
ring,  viz.,  acd — a^c'd',  the  upper  one;  ef  ffh — e"f^  ifh'^  the  middle 
one;  and  rjkl — r'fk'l\  the  lower  one. 

The  circles  of  the  plan  are  found  by  projecting  the  points  as/?' 
upon  the  diameter  AC.  Then  by  comparing  kk',  IV,  and  nn\  for 
example,  we  see  how  the  vertical  projections,  as  k'l'n\  of  the  ends 
of  the  half  rings  are  found. 

loY.  Execution. — The  section  lines  in  the  elevation  indicate 
clearly  the  situation  of  the  three  segments,  ahcd^  €,fgh^  and  ijkl^ 
there  shown.  The  dark  bands  on  the  case  at  t  and  t\  indicate  the 
inlaid  bands  of  soft  metal  already  described. 

The  studt'nt  can  usefully  multiply  these  examples  by  constructing 
pla?is,  elevations  and  sections,  i'rom  measurement,  of  such  objects 
as  steam,  water,  and  gas  cocks,  valves,  or  gates,  raihoay  joints, 
and  chairs,  and  other  like  simple  metallic  details;  actual  exam])lee 
of  which  can  be  easily  obtained  as  models  almost  anywhere. 

Ex.  7.     Rolled  Iron  Beams  and  Columns. 

Wood  is  peri.shalile  and  growing  scarce.  Stone  is  comparatively 
costly  and  cumbrous.     Cast  iron  is  suspected  as  treacherous. 


Fig.  1. 

Hence,  of  late  years,  much  attention  Las  l)ecn  paid  tol)oams  and 
colurauu  of  rolled  wrought  iron.  Various  figures  of  such  work 
arc  therefore  given  as  an  appropriate  concluding  general  example 
for  the  ])rcsont  chajjter. 


PL.VIll. 


COXSTRUCTIOXS    IX    MKTAL. 


61 


Rolled  iron  in  its  elementary  commercial  forms,  for  architec- 
tural and  engineering  purposes,  is  principally  known  as  beam, 
plate,  angle,  T  (Fig.  1),  V  (Fig.  2),  and  chaimel  iron,  the  latter 


Fvj.  3. 


■||'^"''M 


Fuj.  5. 

either  in  polygonal  or  circular  segments  (Fig.  3).     All  of  these 
are  used  in  building  up  compound  beams,  braces,  or  columns. 
Figs.  4-10,  of  those  here  given,*  may  all  be  taken  as  on  a  scale 


1  ij.  u. 


Fig.  7. 


of  one  fourth  the  full  size,  and  Fig.   9,  as  a  practical  example 
of  oblique  projection  (see  Div.  IV.),  of  one  eighth  the  full  size. 


*  From  the  Uniou  Iron  Works,  Buffalo,  N.  Y. 


63 


CONSTRUCTIONS   IN    METAL. 


Figs.  4-13;,  excciot  9,  are  all  transverse  sections  oi  the  beams  or 
columns  which  they  represent. 


..  \_ 


Fig.  8, 

Simple  beams  being  made  of  all  sizes  from  four  io  fifteen  inches 
in  depth,  and  some  of  the  sizes  either  light  or  heavy,  Figs.  4-8 
represent  compound  beams  of  depths  greater  than  fifteen  inches. 


Fig.  4  represents  the  lower  half  of  a  beam  formed  of  riveted 
plate  and  angle  irons. 


CONSTRUCTIONS   IX    METAL. 


63 


Fkj.  10. 


Fig.  11. 


Fig.  12. 


64  CONSTRUCTIOXS   IN    METAL. 

Fig.  5  represents  the  left-hand  half  of  a  holloTV  beam  of  hori- 
zontal plate,  and  Tertical  rectangular  channel  iron. 

Fig.  6  shows  the  lower  end  of  a  very  deep  beam,  the  ends  of 
which  are  symmetrically  placed,  and  formed  of  plate  and  curved 
channel  iron. 

Fig.  7  is  the  lower  half  of  a  beam  composed  of  a  simple  beam 
riveted  to  horizontal  plates. 

Fig.  8  represents  the  lower  half  of  a  beam,  16"  wide  at  base, 
composed  of  plate,  beam,  and  rectangular  channel  irons. 

Fig.  9,  an  oblique  projection,  given  here  for  convenience  in 
anticipation  of  Div.  IV.,  shows  how  beams  at  right  angles  to  each 
other  may  l)e  riveted  together  by  angle  plates. 

Leaving  ])eams  for  the  highly  interesting  and  practically  very 
important  subject  of  iron  columns,  Fig.  10  is  the  partial  hori- 
zontal section  of  a  column  curiously  formed  of  two  flanged  beams, 
bent  at  right  angles,  and  riveted  along  the  angle,  through  an 
intermediate  doubly  concave  bar,  which  gives  a  firmer  bearing 
for  the  rivets. 

Fig.  11  is  nearly  a  half  section  of  a  column  composed  of  six 
curved  channel  irons,  disposed  with  the  convexities  inwards,  and 
riveted  in  the  angles  of  the  flanges. 

Fig.  12  shows  two  examples  of  the  trvie  hollow  column  *  as 
made  in  segments,  consisting  of  channel  irons  whose  fl.anges  are 
radial  and  whicli  are  placed  with  their  convexities  outward,  and 
are  riveted  through  the  flanges.  The  inner  figure  represents  a " 
column  of  four  channel  irons,  the  outer  one  a  larger  column  of  six 
flanges ;  the  number  varying  from  four  to  eight  in  different  cases. 

Fig.  13  is  a  section  of  a  column  of  German  design,  made  of 
plate  and  polygonal  channel  irons. 

The  ideal  of  Fig..  12  is  a  smooth  hollow  column  iu  one  piece, 
a  given  amount  of  matter  having  much  greater  strength,  in  this 
form,  to  resist  crushing  or  bending  than  when  in  a  solid  bar. 
But  this  ideal,  though  easily  realized  in  cast  iron,  is  economical! j 
impracticable  in  wrouglit  iron;  hence  the  external  radial  flanges, 
though  giving  additional  strength  by  increasing  the  average  dis- 
tance of  the  material  from  the  centre,  are  subordinate  to  the 
main  idea. 

In  Fig.  11,  on  tlic  contrary,  the  heavy  flanges  are  thejirimary 
feature  as  a  means  of  gaining  strength  l)y  a  circumferential  dis- 

*  ^ladp  at  Plia>nixville,  Pa. 


CONSTKUCTIONS   IN    METAL. 


65 


position  of  material,  while  the  curved  parts  form  a  meang  of 
connecting  them,  and  of  also  leaving  a  hollow  interior. 

A  plate  alone  easily  "buckles"  in  the  direction  of  its  thick- 
ness. Each  half  of  Fig.  10  is  therefore  a  plate,  braced  to  prevent 
this  by  bending  at  right  angles,  and  by  means  of  flanges  which 
also  make  the  circumferential  strength  prominent. 

Fig.  13  is  also  essentially  a  braced  plate  column,  the  primary 


office  of  the  channgl  irons  bein^j  to  unite  the  plates,  leaving  their 
effect  on  the  circumferential  strength  incidental. 

If  a  column  could  be  formed  by  placing  the  channel  irons  of 
Fig.  11  so  as  to  join  them  by  their  flanges,  which,  in  Fig.  13, 
would  then  radiate  inward  from  the  circumference  of  the  column, 
this  circumference  might  then  be  of  slightly  greater  diameter 
for  a  given  amount  of  matter.  But  the  riveting  would  be 
inaccessible.  It  may  be  a  question,  however, 
whether  collars  might  not  be  clamped  or 
shrunk  on,  numerous  enough  to  make  the 
column  essentially  solid,  and  at  no  more  than 
the  cost  of  riveting.  This  question  appears 
to  have  exercised  other  minds,  and  Fig.  14 
indicates  one  solution  of  it,*  consisting  of 
cylindrical  segments  bearing  dovetailed  in 
jDlace  of  rectangular  flanges  on  their  edges, 
and  united  by  clamping  bars  in  place  of  riv- 
ets ;  as  one  would  hold  two  boards  laid  one 
upon  the  other,  by  clasping  each  hand  over  the 
two  edges  of  the  boards,  while  the  riveting  plan  may  be  repre- 
sented by  thrusting  the  fingers  through  holes  near  the  edges  of 
the  boards. 


*  Manufactured  l)y  Carnegie  Brothers  &  Co.,  Pittsburg,  Pa. 


DIVISION    TIIII113. 

IK 

ELEMENTARY  SHADOWS  AND  SHADING. 


CHAPTER  I. 

SHADOWS. 

§  I, — Fads,  Principles,  and  Preliminary  Problems. 

158.  The  shadow  of  a  given  opaque  body,  B,  PI.  IX,,  Fig. 
78,  upon  any  surJace  S,  is  the  portion  of  S  from  which  hght  is 
excluded  by  B. 

A  shadow  is  known  when  its  boundary,  splc,  called  the  line  of 
shadow,  is  known.  Hence,  to  find  the  shadow  of  a  given  body 
upon  a  given  surface  is,  practically,  to  find  the  boundary  of  that 
shadow, 

159.  The  boundar}^  sph,  PI.  IX,,  Fig,  78,  of  the  shadow  of  a 
body,  B,  is  the  shadow  of  the  line  of  shade,  hrn,  which  divides  the 
illuminated  from  the  unilluminated  part  of  B,  For  if  a  ray,  od, 
pierces  S,  as  at  d,  ivithm  the  area  of  the  shadow,  it  must  pierce  the 
body  B  in  its  illuminated  part  as  at  o;  but  if  another  ray  pierces 
S,  as  at  q,  without  the  shadow,  it  cannot  meet  the  body  B.  Hence 
rays,  as  hs,  cp,  nh,  which  meet  S  in  the  li7ie  of  shadow,  must  be 
tangent  to  B  at  points,  as  ?i,  c,  n,  of  its  line  of  shade. 

160.  Since  the  line  of  shadoio  of  anybody  B  on  any  surface  S 
is  the  shadow  of  the  line  of  shade  of  B,  the  line  of  shade  on  the 
body  casting  a  required  shadow  must,  in  general,  be  found  first 
in  problems  of  shadows. 

On  plane-sided,  that  is  flat-sided  bodies,  this  line  of  shade  con- 
sists simply  of  the  edges  which  divide  its  faces  in  the  light  from 
those  in  the  dark;  that  is,  it  consists  of  the  shade  lines  (21)  of  the 
body.     These  may  in  many  cases  be  found  by  simple  inspection, 

101.  Rays  of  liyht  are  here  assumed  to  he  parallel  straight  lines, 
as  they  practically  are  when  proceeding  from  a  very  distant 
source,  as  the  sun,  to  any  terrestrial  object. 

\st.  It  will  be  thus  observed  that  the  shadow  of  a  vertical  edge 
ab,  PI.  IX.,  Fig.  78,  of  the  body  of  the  house  will  be  a  vertical 
line,  a'h',  on  the  front  wall  of  the  wing  behind  it;  that  the  shadow 


SHADOWS.  G7 

of  a  horizontal  line,  as  he — the  arm  for  a  swinging  sign — which  is 
parallel  to  the  wing  wall,  will  be  a  horizontal  line,  b'e\  parallel 
to  be;  that  a  horizontal  line,  be,  which  is  perpendicular  to  the  wing 
wall,  will  have  an  oblique  shadow,  cb',  on  that  wall,  commencing 
at  c,  where  the  line  pierces  the  wing  wall,  and  ending  at  b',  where 
a  ray  of  light  through  b  pierces  the  wing  wall;  and  finally  that 
the  shadow  of  a  point,  b,  is  at  b'  where  the  ray  bb' ,  through  that 
point,  pierces  the  surface  receiving  the  shadow. 

2c?.  Passing  now  to  PL  IX.,  Fig.  79,  which  represents  a  chim- 
ney upon  a  flat  roof,  we  observe  that  the  shadows  of  he  and  ed— 
lines  parallel  to  the  roof — are  b'c'  and  c'cV,  lines  equal  to,  and 
parallel  to,  the  lines  be  and  cd;  and  that  the  shadows  of  ab  and  ed 
are  ab'  and  ed' — similar  to  the  shadow  eb'  m  Fig.  78 — i.  e.  com- 
mencing at  a  and  e,  where  the  lines  casting  them  meet  the  roof, 
and  ending  at  b'  and  d' ,  where  rays  through  b  and  d  meet  the  roof. 

1G2.  The  facts  just  noted  may  be  stated  as  elementarij  general 
principles  by  means  of  which  many  simple  problems  can  be  solved. 

1st.  The  shadow  of  a  point  on  any  surface,  is  where  a  ray  of 
light  through  that  point  meets  that  surface. 

2rf.  The  shadow  of  a  straight  line,  upon  a  plane  parallel  to  it, 
is  a  parallel  straight  line. 

3f7.  In  like  manner,  the  shadow  of  a  circle  upon  a  plane  parallel 
to  it  is  an  equal  circle  ;  whose  eentre,  only,  therefore  need  be  found. 

A:th.  The  shadow  of  a  line  upon  a  plane  to  which  it  is  perpen- 
dicular, will  coincide  with  the  projection  of  a  ray  of  light  upon  that 
plane.  Thus  ba,  Fig.  78,  being  perpendicular  to  H,  its  shadow 
upon  that  plane  coincides  with  the  horizontal  projection,  aa' ,  of 
the  ray  of  light  bb'.  Likewise  be,  being  perpendicular  to  the 
vertical  plane  cb'a',  its  shadoAV  coincides  with  cb',  the  projection 
of  bb'  on  that  plane. 

bth.  The  shadow  of  a  line  upon  a  surface  may  be  said  to  begin 
where  that  line  meets  that  surface,  either  or  both  being  produced, 
if  necessary.     See  a,  Fig.  79. 

%th.  When  the  shadow,  as  aa'b',  of  a  line,  as  ah.  Fig.  78,  falls 
upon  two  surfaces  which  intersect,  the  partial  shadows,  as  aa' 
and  h'a',  meet  this  intersection  at  the  same  point,  as  a'. 

Ith.  The  shadow  of  a  straight  line,  upon  a  curved  surface,  or 
of  a  circle,  otherwise  than  as  in  (3fZ)  will  generally  be  a  curve, 
whose  points,  separately  found  by  {1st),  must  then  be  joined. 


68  SHADOWS. 

163.  In  applying  these  principles  to  the  construction  of  shad- 
ows, three  things  must  evidently  be  given,  viz. — 

Isf.  The  body  casting  the  shadow. 

2d.  The  surface  receiving  the  shadow. 

3d.  The  direction  of  the  light. 

And  these  must  be  given,  not  in  reality,  but  in  projection. 
i     164,  The  light  is,  for  convenience  and  uniformity,  usually  as- 
sumed to  be  in  such  a  direction  that  its  projections  make  angles  of 
45°  with  the  ground  line  (16).     For  the  direction  of  the  light  it- 
self, corresponding  to  these  projections  of  it,  see  PI.  IX.,  Fig.  80. 

Let  a  cube  be  placed  so  that  one  of  its  faces,  h'hj),  shall  coin- 
cide with  a  vertical  plane,  and  another  face,  L"aLi,  with  the 
horizontal  plane.  The  diagonals,  L'L,  and  L"Li,  of  these  faces, 
will  be  the  projections  of  a  ray  of  light,  and  the  diagonal,  LLi,  of 
the  cube  will  be  the  ray  itself  ;  for  the  point  of  Avhich  L'  and  U' 
are  the  projections  must  be  in  each  of  the  projecting  perpendicu- 
lars L'L'  and  L"L ;  hence  at  L,  their  intersection.  Now  the 
right-angled  triangles,  LL"Li  and  LL'Lj,  are  equal,  but  not  isos- 
celes ;  hence  the  angles  LLjL"  and  LLiL',  which  the  ray  itself, 
LLi,  makes  with  the  planes  of  projection  are  equal,  but  less  than 
45°.  Again:  in  tlie  triangle  LLiG,  the  angle  LLjG  is  that  made 
by  the  ray  LLj  with  the  ground  line,  and  is  77iore  than  45°. 

165.  Having  now  stated  the  elementary  facts  and  principles 
concerning  shadows  themselves,  we  proceed  to  show  how  to  find 
the'iv  jjroject ions,  by  which  they  are  represented. 

By  (162)  we  have  first  to  learn  how  to  find  where  a  given  line, 
as  a  ray  of  light,  pierces  a  given  surface.  It  will  be  sufficient 
here  to  show  how  to  construct  the  points  in  which  a  line  pierces 
the  2:)lanes  of  projection. 

Prelimi7iary  Problems. 

Problem  \st.  To  find  where  a  line  drawn  through  a  given 
point,  pierces  the  horizontal  plane. 

The  construction  is  shown  pictorially  in  Fig.  1,  and  in  actual 
projection  in  Fig.  2. 

Let  A,  Fig.  1,  be  the  given  jioint,  wliose  projections  are  repre- 
sented by  a  and  «'.  If  a  line  ])asscs  through  a  point,  its  projec- 
tions will  pass  through  the  projections  of  that  point;  thoi'efore 
if  AB  be  the  line,  ab  and  a'l/  will  be  its  projections. 

'Sow,  first,  the  7'equired  {nAnt,  C,  being  in  the  horizontal  plane, 


SHADOWS.  09 

its  vertical  projection,  c',  will  be  on  the  ground  line;  second,  the 
same  point  being  a  point  of  the  given  line,  its  vertical  projection 
wil!  be  on  the  vertical  projection,  a'h'c',  of  the  line,  hence  at  the 
intersection,  c',  of  that  projection  with  the  ground  line;  third,  the 
horizontal  ])rojection,  c,  of  the  point,  which  is  also  the  point  itself, 
C,  will  be  where  the  perpendicular  to  the  ground  line  from  c'  meets 
abc,  the  horizontal  projection  of  the  line. 

This  explanation  may  be  condensed  into  the  following  rule. 

l5^  Note  where  the  vertical  ])rojection  of  the  given  line,  pro 
duced  if  necessary,  meets  the  ground  line. 

2d.  Project  this  point  into  the  horizontal  projection  of  the  line; 
which  will  give  the  required  point. 

Fig.  2,  shows  the  application  of  this  rule  in  actual  projection. 
ab — a'b'  is  the  given  line,  and  by  making  the  construction,  we  find 
c,  wliere  it  pierces  the  horizontal  plane. 


Fig.  1. 


Fig.  2. 


Example. — If  bd  (both  figures)  had  been  less  than  b'd,  the  line 
would  have  pierced  the  horizontal  plane  behind  the  ground  line, 
tliat  is  in  the  horizontal  plane  produced  bachoards.  Let  this  con 
etruction  be  made,  both  pictoi'ally,  and  in  projection. 

Pboblem  2d. — To  find  where  a  line,  drmcn  through  a  given  point., 
pierces  the  vertical  plane. 

See  Figs.  3  and  4.  The  explanation  is  entirely  similar  to  the 
preceding.  The  requiied  point,  c',  being  somewhere  in  the  vertical 
plane,  its  horizontal  projection,  c,  will  be  on  the  ground  line.  It 
being  also  on  the  given  line,  its  horizontal  projection  is  on  the 
horizontal  projection,  nhc.  of  the  line.     Hnnce.  nt  once,  che  rule 


70  SHADOWS. 

lat.  Note  where  the  horizontal  projection  of  the  given  line  mreti 
the  ground  line. 


Pio.  3. 


Fia.  4. 


'2d.  Project  this  point  into  the  vertical  projection  of  the  line, 
which  will  give  c'=C,  the  vertical  projection  of  the  point,  which  is 
also  the  point  itself. 

Example. — Here  likewise,  if  b'd  be  ^55  than  bd,  the  line  will 
pierce  the  vertical  plane  below  the  ground  line,  that  is  in  the  verti- 
cal plane  extended  downward.     Let  the  construction  be  made. 

166.  The  ground  line  is,  really,  the  horizontal  trace  of  the  verti- 
cal plane;  also  the  vertical  trace  of  the  horizontal  plane.  Hence 
the  traces  of  any  other  horizontal  or  vertical  planes  may  be  called 
the  ground  lines  of  those  planes ;  and  therefore  the  preceding  con- 
etrnctions  may  be  applied  to  finding  the  shadows  on  such  planes, 
as  will  shortly  be  seen  in  the  solution  ot"  problems. 

107.  In  respect  to  the  remaining  principles  of  (162)  it  only  re- 
mains to  note  that,  as  two  points  determine  a  straight  line,  it  is 
sufficient  to  find  the  shadow  of  two  jooints  of  a  straight  line,  when 
its  shadow  falls  on  a  plane.  But  if  the  direction,  of  the  shadow  is 
known  as  in  (1G2 — 2f?,  \th)  or  if  we  know  where  the  line  meets  a 
plane  surface  receiving  the  shadow,  (162 — hth)  it  will  be  suflScient 
to  construct  one  point  of  the  shadow. 

§  II. — Practical  Problems. 
168.  FnoB.  1.     To  find  the  shadow  of  a  vertical  beam.,  upon  a 
i^tical  v:all.      PI.   IX.,  Fig.  81.     Let  AA' be  the  beam,  let  the 


SHADOWS.  71 

vertical  plane  of  projection  be  taken  as  tlie  vertical  wall,  and  let  the 
light  be  indicated  by  the  lines,  as  ah — a"b'.  The  edges  wliich  cast 
the  visible  shadow  are  cf. — a'a"  ;  ac — a"/  and  ce — a"e'.  The  sha- 
dow of  a — a'a''  is  a  vertical  line  from  the  point  b\  which  point  is 
where  the  ray  from  a — a"  pierces  the  vertical  plane,  ab — a"b' 
pierces  the  vertical  plane  in  a  point  whose  horizontal  pi-qjection  ia 
b.  b'  must  be  in  a  perpendicular  to  the  groimd  line  fioni  b  (Art.  ]  6), 
and  also  in  the  vertical  projection,  a"b\  of  the  ray,  hence  at  i'.  Ub 
is  therefore  the  shadow  of  a— a^'a'.  The  shadow  of  ac — a"  is  the 
line  b'd\  limited  by  d\  the  shadow  of  the  point  c — a"  (162).  The 
shadow  of  ce — a"e,'  begins  at  d\  and  is  parallel  to  ce — a"e\  but  is 
partly  hidden. 

169.  Execution. — The  bonndary  of  a  shadow  being  determined, 
its  surfice  is,  in  practice,  indicated  by  shading,  either  with  a  tint  of 
indian  ink,  or  by  parallel  shade  lines.  The  latter  method,  affording 
useful  pen  practice,  may  be  profitably  adopted. 

170.  Prob.  2.  To  find  the  shadow  of  an  oblique  timber,  lohich  is 
parallel  to  the  vertical  plane,  upon  a  similar  timber  resting  against 
the  back  of  it.  PI.  IX.,  Fig.  82.  Let  AA'  be  the  timber  which 
casts  a  shadow  on  BB',  which  slants  in  an  opposite  direction.  The 
edgeac — aV  of  A  A',  casts  a  shadow,  parallel  to  itself,  on  the  front 
face  of  BB';  hence  but  one  point  of  this  shadow  need  be  con- 
structed. Two,  however,  are  found,  one  being  a  check  upon  the 
other.  Any  point,  aa\  taken  at  pleasure  in  the  edge  ac — aV  casts 
a  point  of  shadow  on  the  front  plane  of  BB',  whose  horizontal 
projection  is  J,  and  whose  vertical  projection  (see  Prob.  1)  must  be 
in  the  ray  a'b\  and  in  a  perpendicular  to  the  ground  line  at  b ; 
hence  it  is  at  b'.  The  shadow  of  ac — a'c'  being  parallel  to  that 
line,  b'd'  is  the  line  of  shadow.  d\  the  shadow  of  c — c\  was  found 
in  a  similar  manner  to  that  jnst  described. 

It  makes  no  difference  that  b'  is  not  on  the  actual  timber,  BB'; 
for  the  face  of  that  timber  is  but  a  limited  physical  plane,  forming 
a  portion  of  the  indefinite  immaterial  plane,  in  which  bb'  is  found  ; 
hence  the  point  b'  is  as  good  for  finding  the  direction  of  the  indefi- 
nite line  of  shadow,  b'd\  as  is  d\  on  the  timber  BB',  for  finding  the 
real  portion  only  of  the  line  of  shadow,  viz.  the  part  which  lies 
across  BB'.    Here,  bd  is  taken  as  a  ground  line  (166). 

Observe  that  the  shade  line  Ab,  of  the  timber  AA'  should  end 
at  bb',  where  the  timbers  intersect. 

Example. — Make  the  timbers  larger,  A  A'  more  nearly  hori- 
zontal than  vertical,  and  then,  in  both  figures,  find  the  shadow 
on  the  plan. 


72  SHADOWS. 

171.  Prob.  3.  To  find  the  shacloxo  of  a  fragment  of  a  horiT/m- 
tal  timber,  vpon  the  horizontal  top  of  an  abutment  on  which  tht 
timber  rests.  PI.  IX.,  Fig.  83.  Let  AA'  be  the  timber,  and  BB' 
the  abutment.  The  vertical  edge,  a — «"a',  casts  a  shadow,  a6,  in 
the  direction  of  the  horizontal  projection  of  a  ray  (102,  4^/f.),  and 
limited  by  the  shadow  of  the  point  a — a'.  The  shadow  of  a — a' is 
at  bb\  where  ttie  ray  ab — a'b'  pierces  the  top  of  the  abutment ;  b' 
being  evidently  the  vertical  projection  of  this  point,  and  b  being 
both  in  a  jierpendicular,  b'b^  to  the  ground  line  and  in  ab,  the  hori- 
zontal projection  of  the  ray.  The  shadow  o^  ad — a'  is  J5",  parallel 
to  ad — a\  and  limited  by  the  ray  a'b' — db".  The  shadow  of 
de — aV  is  b"c^  parallel  to  de — a'e'^  and  limited  by  the  edge  of  the 
abutment  (162,  2d).     Here,  a"b'  is  used  as  a  ground  line  (166). 

172.  Pkob.  4.  To  find  the  shadow  of  an  oblique  timber.,  upon  a 
horizontal  timber  into  xchich  it  is  framed.  PL  IX.,  F'ig.  84. 
The  upper  back  edge,  ca — c'a',  and  the  lower  front  edge  through 
ee',  of  the  oblique  piece,  are  those  which  cast  shadows.  By  con- 
sidering the  point,  c,  in  the  shadow  of  be,  Fig.  78,  it  appears  that 
the  shadow  of  ac — aV,  Fig.  84,  begins  at  cc\  where  that  edge 
pierces  the  upper  sui-faco  of  the  timber,  BB',  which  receives  the 
shadow.  Any  other  point,  as  aa',  casts  a  shadow,  bb%  on  the  plane 
of  the  upper  suifxce  of  BB',  whose  vertical  projection  is  evidently 
b',  the  intersection  of  the  vertical  ))rojection,  a'b\  of  the  ray  at — a'b' 
and  the  vertical  projection,  eV,  of  the  upper  surface  of  BB',  and  whose 
horizontal  projection,  b,  must  be  in  a  projecting  line,  b'b,  and  in  the 
horizontal  projection,  ab,  of  the  ray.  Likewise  the  line  through  ee', 
and  parallel  to  c5,  is  the  shadow  of  the  lower  front  edge  of  the 
oblique  timber  upon  the  top  of  BB'.  This  shadow  is  real,  only  so 
far  as  it  is  actually  on  the  top  surface  of  BB',  and  is  visible  and 
therefore  shaded,  only  where  not  hidden  by  the  oblique  piece. 
Where  thus  hidden,  its  boundary  is  dotted,  as  shown  at  ea.  The 
point,  bb',  is  in  th«  plane  of  the  top  surface  of  BW,  produced. 

Remark. — Thus  it  appears  that  when  a  line  is  oblique  to  a  plane 
containing  its  shadow,  the  direction  of  the  shadow  is  unknown  till 
found.     Let  this  and  the  Ibllowing  iigures  be  made  much  larger. 

Examim.es. — l,s/.  Find  the  shadow  when  the  oblique  timber  is 
more  nearly  vertical  tlian  horizontal. 

2d.  Let  the  oblique  timber  ascend  to  the  right. 

1 73.  Pkou.  5.  To  find  the  shadoio  of  the  side  waU  of  a  flight  of 
tteps  upon  the  faces  of  the  steps.  PI.  IX.,  Fig.  85.  The  stepi 
can   be  easily  constructed  in  good   pro])ortion,  without  measure 


SIIADOAVS.  73 

mcnts,  by  making  the  height  of  each  step  two-thirds  of  its  width, 
taking  four  steps,  and  making  the  piers  rectangular  prisms. 

The  edges,  aa" — a'  and  a — ra',  of  the  left  hand  side  wall  are 
those  which  cast  shadows  on  the  steps, 

Tlie  former  line  casts  horizontal  shadows,  as  hh" — h\  parallel  to 
iliself,  on  the  ?o/as  of  the  steps  (162,  2c?),  and  shadows,  as  he" — h'c\ 
on  the  fronts  of  the  steps,  in  tlie  direction  of  the  vertical  jirojectioii, 
a'd\  of  a  ray  of  light  (162,  ^th) — from  the  upper  step  down  to  tlie 
shadow  of  the  point  aa' .  Likewise,  the  edge  a — ra'  casts  vertical 
shadows,  as  g — g'h\  on  \\\(i  fronts  of  the  steps  (161),  and  shadows 
on  their  tops,  parallel  to  ac?,  the  horizontal  projection  of  a  ray 
(IG'2,  Mli)  from  the  lower  step,  iip  to  (7^^,  the  shadow  o'i  aa' ;  which 
is  tlierefore,  where  the  shadows  of  aa" — a',  and  a — ra',  meet,  a'd' 
is  the  vertical  projection  of  all  rays  thi'ougii  points  of  aa" — a' j 
hence  project  down  h',  etc.,  to  find  the  parallel  shadows,  b"b,  etc. 
Likewise  project  up  g^  etc.,  to  find  the  shadows,  ^7/,  etc.,  ac?  being 
the  horizontal  projection  of  all  rays  through  points  of  a — ra'. 

ExAMPi>ES. —  \st.  Vary  the  jiroportions  of  the  ste2ys  and  the  direc- 
tion of  the  light/  and  in  each  case,  find  the  shadows,  as  above. 

2d.  Bj  pj-eUmi?mr7/  problems  1st  and  2d,  find  directly  where 
the  ray  through  aa'  pierces  the  steps,  only  remembering  that  the 
projections,  d  and  d',  must  be  on  the  same  surface. 

3c?.  Let  the  piers  be  cut  off  by  a  plane  parallel  to  that  of  the 
front  edges  of  the  steps.  (Use  an  end  elevation.) 

174.  Prob.  6.  To  find  the  shadoxo  of  a  short  cylinder^  or  washer^ 
ripon  the  vertical  face  of  a  board.  PL  IX..  Fig.  86.  Since  the 
circidar  face  of  the  washer  is  i)arallel  to  the  vertical  face  of  the 
board,  BB',  its  shadow  will  be  an   equal  circle  (161,  8c?),  of  which 

.we  have  only  to  find  the  centre,  00'.  Tiiis  point  will  be  the  sha- 
dow of  the  point  CC  of  the  washer,  and  is  where  the  ray  CO — CO' 
pierces  the  board  BB'.  The  elements,  rv — r'  and  tu — t'.,  which 
separate  the  light  and  shade  of  the  cylindrical  surface,  have  the 
tangents  r'r"  and  t't"  for  their  shadows.  These  tangents,  with 
the  semicircle  t"r'\  make  the  complete  outline  of  the  required 
shadow. 

175.  Pbob.  7.  To  find  the  shadow  of  a  nut.,  iipoyi  a  vertical 
sinface,  the  nut  having  any  j^ositioji.  PI.  IX.,  P^ig.  86.  Lefc 
a'c'e' — ace  be  the  projections  of  the  mit,  and  BB'  the  projections 
of  the  surface  receiving  the  shadow.  The  edges,  a'c' — ac  and 
ce — c'e',  of  the  nut  cast  shadows  parallel  to  themselves,  since  they 
are  parallel  to  the  surface  which  receives  the  shadow,     a'n' — an 


74  SHADOWS. 

are  the  two  projections  of  the  ray  whicli  determines  the  joint  o\ 
shadow,  wn'/  cV — co  are  the  projections  of  the  ray  used  in  lind 
ing  oo',  and  eV — er  is  the  ray  which  gives  the  point  of  shadow 
n^.  The  edges  at  aa'  and  ee%  whicli  are  perpendicular  to  BB',  cms; 
shadows  a'n'  au<\  e'/,  in  the  direction  of  tlie  projection  of  a  ray  of 
light  on  BB'.     (See  cb\  the  shadow  of  cb,  PI.  IX.,  Fig.  78.) 

176.  PuoB.  8.  To  Jin d  the  shadow  of  a  vertical  cylinder^  on  a 
vertical  plane.  PI.  IX.,  Fig.  87.  The  lines  of  the  cylinder,  CC, 
which  cast  visible  shadows,  are  the  element  a — a"a\  to  which  the 
rays  of  light  are  tangent,  and  a  part  of  the  upper  base.  The 
shadow  of  a — a."a\  is  gg\  found  by  the  m:>thod  given  in  Prob.  I. 
At  g\  the  curved  shadow  of  the  upper  base  begins.  This  is  found 
by  means  of  the  shadows  of  several  points,  hd\  cc\  dd\  &c.  Each 
of  these  points  of  shadow  is  found  as  g  was,  and  then  they  are 
connected  by  hand,  or  l)y  the  aid  of  the  curved  ruler. 

It  is  well  to  construct  one  invisible  point,  as  ?/,  of  the  shadow,  to 
assist  in  locating  more  accurately  the  visible  portion  of  the  curved 
shadow  line. 

177.  Prob.  9.  To  find  the  shadow  of  a  horizontal  beam^  upon 
the  slanting  face  of  an  oblique  abutment.  PI.  IX.,  Fig.  88.  The 
simple  facts  illustrated  l)y  Pl.  IX.,  Figs.  78-79,  have  no  reference 
to  the  case  of  surfaces  of  shadow,  other  than  vertical  or  horizontal. 
But  they  illustrate  the  fact  that  the  point  where  a  shadow,  as  aa\ 
PI.  IX.,  Fig.  78,  on  one  sui'face, meets  another  surface,  is  a  point 
{a')  of  the  shadow  a'b'  upon  that  surface.  Tims  this  problem  may 
be  solved  in  an  elementary  inanner  by  proceeding  indirectly,  i.  e., 
by  finding  the  shadoAvs  on  the  horizontal  top  of  the  abutment  and 
on  its  horizontal  base.  The  points  where  these  shadows  meet  the 
front  edges  of  this  top  and  this  base,  will  be  points  of  the  shadow 
on  the  slanting  face,  que. 

By  Prob.  3  is  Ibuiid  gc^  the  shadow  of  the  upper  back  edge, 
ax — a'x\  of  the  timber,  AA',  upon  the  top  of  the  abutment,  c,  the 
point  where  it  meets  the  front  edge,  cc,  is  a  point  of  the  shadow  ol 
AA'  on  the  inclined  face.  By  a  similar  construction  with  any  ray, 
as  b2y — b'p\  IS  found  qp^  the  shadow  of  c'b' — db  upon  the  base  of 
the  abutment;  and  2',  where  it  intersects  Jig-,  is  a  j)ointof  the  shadow 
of  db — c'b'  on  the  face,  qne.  The  point,  dd\  where  the  edge, 
db — c'J',  meets  f?c,  is  another  point  of  the  shadow  of  that  edge; 
hence  dq — d'q'  is  the  shadow  of  the  fiont  lower  edge,  db — c'b\  on 
the  inclined  face  of  the  abutment.  The  line  through  cc\  parallel  to 
^.q — d'q\  is  the  shadow  of  the  upper  back  edge,  ax — aV,  and  com- 
tes  the  solution. 


SHADOWS. 


75 


178.  Prob.  10.  To  find  the  shadoxo  of  a  pair  of  horizontal  tim- 
bers^ which  are  inclined  to  the  vertical  plane,  ujyon  that  plane.  P!. 
IX.,  Fig.  89.  Let  the  given  bodies  be  situated  as  shown  in  the 
dingram.  In  the  elevation  we  see  the  thickness  of  one  tiinl)er  only, 
because  the  two  limbers  are  supposed  to  be  of  equal  thickness  and 
halved  together.  As  neither  of  the  pieces  is  either  ])arallel  or 
perpendicular  to  the  vertical  plane,  we  do  not  know,  in  advance, 
the  direction  of  their  shadows.  It  will  therefore  be  necessary  to 
find  the  shadows  of  two  points  of  one  edge,  and  one,  of  the  diago- 
nally opposite  edge,  of  each  timber.  The  edges  which  cast  shadows 
are  ac — a'c'  and  ht — e'h\  of  one  timber,  and  ed — e'k  and  niv—a'm" 
of  the  other.  All  this  being  understood,  it  will  be  enough  to  point 
out  the  shadows  of  the  required  points,  without  describing  their 
construction.  (See  hb'.  Fig.  81.)  bb'  is  the  shadow  of  aa',  and  dd^ 
is  the  shadow  of  cc'  y'  hence  the  shadow  line  b^ — d  is  determined. 
So  uu'  is  the  shadow  of  hh' ;  hence  the  shadow  line  u'w'  may  be 
drawn  parallel  to  b'd'.  Similarly,  for  the  other  timber,  ff  is  the 
shadow  of  ee',  and  oo'of  mm\  One  other  point  is  necessary,  which 
the  student  can  construct.  The  process  might  be  shortened  some- 
what by  finding  the  shadows  of  the  points  of  intersection,/?  and  r, 
which  would  have  been  the  points/*"  and  /'  of  the  intersection  of 
the  shadows,  and  thus,  points  common  to  both  shadows. 

i79.  Prob.  11.  To  find  the  shadow  of  a  pair  of  horizontal  tini' 
Ocrs,  which  intersect  as  in  the  last  problem,  up>on  the  inclined  fact 
of  an  oblique  abutment.  PI.  IX.,  Fig.  90.  Tiiis  i:>roblem  is  so 
similar  to  Prob.  9,  that  we  only  note,  as  a  key  to  it,  that  sd,  corres- 
ponding to  2)q,  Fig.  88,  is  the  auxiliary  shadow  of  the  edge,  Ab — n'b', 
upon  a  horizontal  plane  s'd',  cutting  the  line  sb  from  the  face  C  of 
the  abutment,  and  thence  giving  a  point  ss',  intersection  of  sd, 
parallel  to  Ab,  with  sb  of  the  shadow  of  A6 — n'b'  on  C. 

The  folloAving  examples  of  the  very  useful  method  of  auxiliary 
elutdoivs  (162,  6th),  and  Prob.  9,  are  here  briefly  added: 

Examples.  -1st.  To  find  the  shadow  of  an  abacus  of  any  form, 
upon  a  conical  column.  PL  IX.,  Fig.  88a.  Circles  OA  and  OB, 
with  A'B'G'  represent  the  conical  pillar;  and  circle  OC,  with  C'D'B', 
its  cylindrical  cap,  or  abacus.  Then  P'Q'  is  a  horizontal  i)lane, 
cutting  from  the  pillar  the  circle  P'Q',  of  radius  OB=:nP;  and 
pierced  by  the  ray  Oa — O'a'  at  a'a  y  centre  of  the  cii'cle,  of  radius 
Od=^0"D' .  Then  (/,  projected  ate/',  and  intersection  of  the  circle 
Od  with  the  circle  OQ,  is  a  point  of  the  shadow  of  circle  OC  — O'D' 
of  the  abacus,  upon  the  pillar ;  since  the  circle  Oc£  is  the  auxiliary 


«"  SHADOWS. 

Bhadow  of  the  circle  OC  —  (YD'  upon  the  plane  P'Q'  (IG'2,  Sff).  OtKei 
points  may  be  likewise  found, 

2d.  7h  find  the  shadow  of  the  front  circle  of  a  niche,  upon  its 
own  sjjherical  surface.  PL  IX.,  Fig.  88b.  ABC  — A'D'C  is  the 
quarter  sphere,-  which  surmonts  the  vertical  cylindrical  part  of  the 
niche  below  the  line  ABC —  A'O'C.  Then  the  ray  Oa  — O'a'  meets 
any  plane  E.S,  parallel  to  the  face,  AOC  —  A'D'C,  of  the  niche,  at 
aa'  y  giving  circle  a',7iin'=  circle  O'A',  for  the  auxiliary  shadow  of 
circle  O'A'  upon  the  plane  KS  (102,  3d),  circle  a',  m'n'  then  cuts 
circle  RS — E'ci'S',  cut  from  the  spherical  part  of  the  niche  by  the 
plane  RS,  at  d' d' ,  a  required  point  of  shadow.  Find  other  j)ointa 
likewise,  and  join  them.     The  tangent  ray  at  i  gives  that  point. 

3d.  To  firul  the  shculov}  of  a  verticcd  staff,  upon  a  hemi  spherical 
dome.  PI.  IX.,  Fig.  88c.  Circle  OC,  with  C'P'D'  is  the  hemi- 
s])here,  and  A— A'B'  the  staff.  By  (162,  ith)  Ae  the  horizontal 
projection  of  a  ray,  is  also  the  shadow  of  A  —  A'B'  upon  the  assumed 
horizontal  plane  P'Q',  and  cutting  from  the  hemisphere  the  circle 
P'Q'— PQ.  Then  dd',  intersection  of  Ae  with  cii-cle  PQ  — P'Q',  is 
one  i^OLiit  of  the  required  shadow  of  A — A'B'  upon  the  dome. 

180.  PnoB.  12.  To  find  the  shadoio  of  the  floor  of  a  bridge 
upon  a  verticcd  cylindrical  ahutment.  PI.  IX.,  Fig.  91.  The 
line  aq — a'g'  is  the  edge  of  the  floor  which  casts  the  shadow. 
hdg—h'e'g'n  is  the  concave  vertical  abutment  receiving  the  shadow. 
gg\  where  the  edge  ag — g'a'  of  the  floor  meets  this  curved  wall,  is 
one  point  of  the  shadow,  /'is  thehcrizont:d  projection  of  the  point 
where  the  ray,  ef — ef\  meets  the  al)utment ;  its  vertical  projection 
is  in  ef,  tlie  vertical  projection  of  the  ray,  ami  in  a  peii)ondicular 
to  the  ground  line,  thi'ough  f  hence  at/"'.  Similnrly  we  find  the 
points  of  shadoA\',  errand  bb',  and  joining  them  with /"'and  g',  have 
the  boundary  of  the  required  shadow.  Observe,  that  to  find  the 
shadow  on  any  ])articular  vertical  line,  as  b — I/'b',  we  draw  the  ray 
in  the  direction  b — a  ;  then  project  a  at  a',  &c. 

Hemark. — The  student  may  profitably  exercise  himself  in  chang- 
ing the  ])Ositions  of  the  given  j^arts,  while  ret.niiiing  the  methods 
cf  solution  now  given. 

For  example,  let  the  parts  of  the  last  problem  be  placed  side 
by  side,  as  two  elevations,  giving  the  shadow  of  a  vertical  wall 
on  a  horizontal  concave  cylindrical  surface;  or,  let  tiie  tinilicrs,  Fig. 
89,  be  in  vertical  planes,  and  let  their  shadows  then  be  found  on  a 
horizontal  surface. 


i 


A 


CHAPTER  II. 

SHADING. 

131.  The  distinction  between  a  shade  and  a  shadotu  is  this. 
A  shadow,  as  indicated  by  the  preceding  problems,  is  the  portion 
of  a  body  from  which  light  is  excluded  by  some  other  'body.  A 
shade,  is  that  portion  of  the  surface  of  a  body  from  Avhich  light 
is  excluded  by  the  body  itself  (158,  159). 

The  accurate  representation  of  shades  assists  in  judging  of  the 
forms  of  bodies  ;  that  of  shadoios  is  similarly  useful,  besides  aid- 
ing in  showing  their  relations  io  surrounding  bodies. 

In  either  case,  a,  flat  tint  mainly  shows  only  ivhere  the  shade, 
or  shadow,  is  ;  while  finished  shading  helps  to  show  the  forin  and 
position  of  the  body. 

182.  Example  1°.    To  shade  the  elevation  of  a  vertical  right  hex^ 
agonal  prism,  and  its  shadow  on  the  horizontal  plane.  PI.  X.,  Figs. 
'.)2  and  A.     Let  the  prism  be  placed  as  represented,  at  some  dis- 
tance from  the  vertical  plane,  and  with  none  of  its  vertical  fac-ea 
parallel  to  the  vertical  plane.     The  face,  A,  of  the  prism  is  in  tho 
light ;  in  fact,  the  light  strikes  it  nearly  perpendicularly,  as  may  be 
seen  by  reference  to  the  plan  ;  hence  it  should  receive  a  very  light 
tint  of  Indian  ink.     The  left  hand  portion  of  the  face,  A,  is  made 
slightly  darker  than  the  right  hand  part,  it  being  more  distant; 
for  the  reflected  rays,  which  reach  us  from  the  left  hand  portion, 
have  to  traverse  a  greater  extent  of  air  than  those  fiom  the  neigh- 
borhood of  the  line  tt',  and  hence  are  more  absorbed  or  retarded; 
since  the  atmosphere  is  not  perfectly  transparent.     That  is,  these 
rays  make  a  weaker  impression  on  the  eye,  causing  the  left  hand 
portion  of  A,  from  which  they  come,  to  appear  darker  than  at  tt' 

Remarks. — a.  It  should  be  remembered  that  the  whole  of 
face  A  is  very  light,  and  the  difference  in  tint  between  its  opposite 
sides  very  slight. 

b.  As  coi'ollai'ies  from  the  preceding,  it  appears :  Jirst^  that  a  sur- 
face parallel  to  the  vertical  plane  would  receive  a  uniform  tint 
throughout ;  and,  second^  that  of  a  series  of  such  surfaces,  all   of 


78  SHADING 

which  are  in  the  light,  tlie  one  nearest  the  eye  would  be  Hghtest, 
and  tlie  one  fuithest  from  the  eye,  darkest. 

c.  It  is  only  for  great  diiferences  in  distance  that  the  above  effects 
are  manifest  in  nature;  but  drawing  by  j)rojections  being  artificial, 
bi)th  in  respect  to  the  shapes  which  it  gives  in  the  drawings,  and  in 
the  abscnc  •  of  snrioiindlng  objects  which  it  allows,  we  are  obliged 
to  exaggerate  natural  appearances  in  some  respects,  in  order  to 
convey  a  clearei-  idea  of  the  forms  of  bodies. 

d.  The  mere  manual  process  of  shading  small  surfaces  is  hero 
briefly  described.  With  a  sharp-pointed  camel's-hair  brush,  wet 
with  a  very  light  tint  of  Indian  ink,  make  a  narrow  strip  against  the 
left  hand  line  of  A,  and  soften  off  its  edge  with  another  brush 
slightly  wet  with  clear  water.  "When  all  this  is  dry,  commence  at 
the  same  line,  and  make  a  similar  but  wider  strip,  and  so  proceed 
till  the  -whole  of  face  A  is  completed,  when  any  little  irregularities 
in  the  gradation  of  shade  can  be  evened  up  with  a  delicate  sable 
brush,  dam2)  only  with  very  light  ink. 

183.  Passing  now  1o  face  B,  it  is,  as  a  whole,  a  little  darker  than 
A,  because,  as  may  be  seen  by  reference  to  the  ])]an,  while  a  beam 
of  rays  of  the  thickness  njy  strikes  face  A,  a  beam  having  only  a  thick 
nesspr,  stiikes  liice  B  ;  i.  e.,  we  assume, _/?rs?,  tliat  the  actual  hrhjht- 
ness  of  a  flat  surface  is  proportioned  to  the  number  of  rays  of  light 
which  it  receives;  and,  second^  that  its  apparent  brightness  is,  other 
things  being  the  same,  proportioned  to  its  actual  brightness.  Also 
the  part  at  a — a'  being  a  little  more  remote  than  the  Hue  t — 1\  the 
part  at  a — a'  is  made  a  very  little  darker. 

184.  The  face  C  is  very  dark,  as  it  receives  no  light,  except 
a  small  amount  by  reflection  from  surrounding  objects.  Tliis 
side,  C,  is  darkest  at  the  edge  a — a'  which  is  nearest  to  the 
eye.  This  agrees  with  experience;  for  while  the  shady  side 
of  a  house  near  to  us  appears  in  strong  contrast  with  the 
ilhuninated  side,  the  shady  side  of  a  remote  building  appears 
Bcarcely  darker  than  the  illuminated  side.  This  fact  is  ex])lained 
as  follows:  The  air,  and  particles  floating  in  it,  between  tlie 
eye  and  the  dark  surface,  C,  are  in  the  light,  and  reflect  some 
light  in  a  direction  fiom  the  dark  surface  C  to  the  eye;  and 
as  the  air  is  invisible,  these  reflections  appear  to  come  from 
that  dark  surface.  Now  the  more  distant  that  surface  io,  the 
gieater  will  be  tlie  body  of  illuminated  air  between  it  and  the 
eye,  and  therefore  the  greater  will  be  the  amount  of  light,  appa- 
rently ))rocee;Jing  from  the  surface,  and  its  consequent  apparent 
brightness.     That  is,  the  more  distant  a  surface  in  the  dark  is,  the 


SHADING.  79 

lit,'hter  it  will  appear.  It  may  be  objected  that  this  would  make  o:it 
the  remoter  parts  of  illuminated  surfaces  as  the  lighter  parts.  IJut 
noi  so  ;  for  the  air  is  a  nearly  perfect  transparent  medium,  and  hence 
reflects  but  little,  compared  with  what  it  transmits  to  the  opaque 
body ;  but  being  not  quite  transparent,  it  absorbs  the  reflected  rays 
from  the  distant  body,  in  proportion  to  the  distance  of  that  body, 
making  therefore  the  remoter  portions  darker;  while  t\\Q  very  tceak 
reflections  from  the  shady  side  are  reinforced,  or  replaced, by  more 
of  the  comparatively  stronger  atmospheric  reflections,  in  case  of  the 
remote,  than  in  case  of  the  near  i3art  of  that  shady  side.  Thus  is 
made  out  a  consistent  theory. 

In  relation,  now,  to  the  shadow,  it  will  be  lightest  where  furthest 
from  the  prism,  since  the  atmospheric  reflections  evidently  have  to 
traverse  a  less  depth  of  darkened  air  in  the  vicinity  of  de — d'  than 
near  the  loiccr  base  of  the  prism  at  ahc. 

185.  Ex.  2".  To  shade  the  elevation  of  a  vertical  cylinder.  PI.  X., 
Fig.  93.  Let  the  cylinder  stand  on  the  horizontal  plane.  The  figures 
on  the  elevation  suggest  the  comparative  depth  of  color  between 
the  lines  adjacent  to  the  figures.  The  reasons  for  so  distributing 
the  tints  will  now  be  given.  See  also  Fig.  B. 

The  darkest  part  of  the  figure  may  properly  be  assumed  to  be 
that  to  which  the  rays  of  light  are  tangent;  viz.,  the  vertical  line  at 
tt\  from  which  the  tint  becomes  lighter  in  both  directions. 

The  lightest  line  is  that  which  reflects  most  light  to  the  eye.  Now 
it  is  a  principle  of  optics  that  the  incident  and  the  reflected  i*ay 
make  equal  angles  with  a  perpendicular  to  a  surface.  But  nQ  is  the 
incident  ray  to  the  centre,  and  Qe  the  reflected  ray  from  C  to  the 
eye  (12).  Hence  d.,  which  bisects  ne,  shows  where  the  incident  ray, 
ed,  and  reflected  ray,  dq^  make  equal  angles  with  the  per})endicular 
(normal)  c?C,  to  the  surface.     Hence  d — d'  is  the  lightest  element. 

186.  Remark. — The  question  may  here  arise,  "If  all  the  light 
that  is  reflected  towards  the  eye  is  reflected  from  d — as  it  appears 
to  be — how  can  any  other  point  of  the  body  be  seen  ?"  To  answer 
this  question  requires  a  notice— ^rs^;,  of  the  difierence  between 
polished  and  dull  surfaces ;  and,  second.,  between  the  case  of  light 
coming  xohollyxw  one  direction,  ox  xyrincipally  in  one  direction.  If 
the  cylinder  CC,  considered  as  perfectly  polished,  were  deprived  of 
all  reflected  light  from  the  air  and  surrounding  objects,  the  line  at 
dr—d'  would  reflect  to  the  eye  all  the  light  that  the  body  would 
remit  towards  the  eye,  and  would  appear  as  a  line  of  brilliant  light, 
while  other  parts,  remitting  no  light  whatever,  would  be  totally 


6U  SHADING. 

invisible.  Let  iis  now  suppose  a  reflecting  inodium,  tliough  ua 
imperfect  one,  as  the  atmosphere,  to  be  thrown  arouml  tlie  body. 
By  reflection,  every  part  of  the  body  would  receive  some  liglit  from 
all  direction?,  and  so  would  remit  some  light  to  the  eye,  making  the 
body  visible,  though  faintly  so.  But  no  body  has  a  polish  that  i? 
absolutely  perfect ;  rntlier,  the  great  majority  of  those  met  with  in 
engineering  art  have  entirely  dull  sui'faces.  Now  a  dull  surface, 
greatly  magnified,  may  be  supposed  to  have  a  structure  like  that 
shown  in  PI.  X.,  Fig.  97,  in  which  many  of  the  nsperities  may  be 
supposed  to  have  one  little  facet  each,  so  situated  as  to  remit  to  the 
eye  a  ray  received  by  the  body  directly  from  the  pnncipa)  source 
of  light. 

187.  Having  thus  shown  how  any  object  placed  before  our  eyes 
can  be  seen,  we  may  proceed  with  an  explanation  of  the  distribu- 
tion of  tints,  b  is  midway  between  d  and  t.  At  A,  the  ink  may  be 
diluted,  and  at  e — e\  much  more  diluted,  as  the  gradation  from  a 
faint  tint  at  e  to  absolute  whiteness  at  d  should  be  without  any 
abrupt  transition  anywhere. 

The  beam  of  incident  rays  which  falls  on  the  segment  dn^  is 
broader  than  that  wdiich  strikes  the  equal  segment  de ,'  hence  the 
segment  nd  is,  on  the  elevation,  marked  5,  as  being  the  lightest  band 
which  is  thited  at  all.  The  segment  w,  behig  a  little  more  obliquely 
illuminated,  is  less  bright,  and  in  elevation  is  marked  3,  and  may  be 
made  darkest  at  the  left  hand  limit.  Finally,  the  segment  ru  receives- 
about  as  much  light  as  r?^,  but  reflects  it  within  the  very  narrow 
limits,  5,  hence  appears  brighter.  This  condensed  beam,  s,  of 
reflected  rays  would  make  rytlie  lightest  band  on  the  cylinder,  but 
for  two  reasons;  Jirst.,  on  account  of  the  exaggerated  eflfect  allowed 
to  increase  of  distance  from  the  eye;  and,  second^  because  some  of 
the  asperities,  Fig.  97,  would  obscui'e  some  of  the  reflected  I'ays 
from  asperities  still  more  remote;  hence  rv  is,  in  elevation,  marked 
4,  and  should  be  darkest  at  its  right  hand  limit. 

188.  The  process  of  shading  is  the  same  as  in  the  last  exercise. 
Each  stripe  of  the  preliminary  ])rocess  may  extend  past  the  i)reccd- 
ing  one,  a  distance  equal  to  that  indicated  by  the  short  dashes  at 
the  top  of  Fig.  94.  When  the  whole  is  finished,  there  should  be  a 
uniform  gradation  of  shade  froni  the  darkest  to  the  lightest  line, 
free  from  all  sudden  transitions  and  minor  irregularities. 

ISO.  Ex.  3°.  To  shade  a  rUjlit  cone  standing  upon  the  horizon- 
lal  plane,  together  with  its  shadow.  Pi.  X.,  Fig.  95.  The  sha- 
dow of  the  cone  on  the  horizontal  plane,  will  evidently  be  bounded 


SHADING. 


81 


by  the  shiidows  of  those  Vines  of  the  convex  surface,  at  which  the 
li<:;ht  is  tangent.  The  vertex  is  coninion  to  both  these  lines,  and 
casts  a  shadow,  y'".  The  sliadows  being  cast  by  strnight  lines  of 
the  conic  surface,  are  straiglit,  and  their  extremities  must  be  in  the 
base,  being  cast  by  lines  of  the  cone,  which  meet  the  horizontal 
plane  in  the  cone's  base;  hence  the  tangents  v"'t  and  v"'t"  are  tlu 
boundaries  of  the  cone's  shadow  on  the  horizontal  plane,  and  tlie 
lines  joining  i' and  i!"  with  the  vertex  aie  the  lines  to  whicli  th( 
rays  of  light  are  tangent;  i.e.,  they  are  the  darkest  lines  of  the 
sliading  ;  hence  tv,  the  visible  one  in  elevation,  is  to  be  vertically 
projected  at  t'v'. 

The  lightest  line  passes  from  vv'  to  the  middle  point,  y,  between 
ti  and  p  in  the  base.  At  q  and  at/)  a  change  in  the  darkness  of  the 
tint  is  made,  as  indicated  by  the  figures  seen  in  the  elevation.  In 
the  case  of  the  cone,  it  will  be  observed  that  the  various  bands  of 
color  are  triangular  rather  than  rectangular,  as  in  tlie  cylinder ;  so 
that  great  care  must  be  taken  to  avoid  tilling  up  the  whole  of  the 
upper  part  of  the  elevation  with  a  dark  shade.  See  PI.  X.,  Fig.  C. 

190.  Ex.  4°.  To  shade  f/te  elevation  of  a  sphere.  PI.  X.,  Fig. 
96.  It  is  evident  that,  if  there  be  a  system  of  pai-allel  rays,  tan- 
gent to  a  s])here,  their  points  of  contact  will  form  a  great  circle, 
perpendicular  to  these  tangents;  and  which  will  divide  the  light 
from  the  shade  of  the  sphere.  That  is,  it  will  be  its  circle  of 
shade.  Each  point  of  this  circle  is  thus  the  point  of  contact  of  one 
tangent  ray  of  light.  If  now,  parallel  planes  of  rays,  that  is,  planes 
l)arallcl  to  the  light,  be  passed  thi'ough  the  sphere,  each  of  thera 
will  cut  a  circle,  great  or  small,  from  the  sphere,  and  there  will  be 
two  rays  tangent  to  it  on  opposite  sides,  whose  jwiiUs  of  contact 
will  be  points  of  the  circle  of  shade. 

In  the  construction,  these  parallel  planes  of  rays  will  be  taken 
perpendicular  to  the  vertical  plane  of  projection. 

Next,  let  us  recolleci,  that  always,  when  a  line  is  parallel  to  a  jjlane, 
rts  projection  on  that  plane  will  be  seen  in  its  true  direction.  Now 
]^Y)'  being  the  direction  of  the  liglit,  as  seen  in  elevation,  let  BD'  bo 
the  trace,  on  the  vertical  plane  of  projection — taken  through  the 
centre  of  the  sphere — of  a  new  plane  perpendicular  to  the  vertica. 
plane,  and  therefore  parallel  to  the  rays  of  light.  The  projection 
of  a  ray  of  light  on  this  plane,  BD',  will  be  pai-allel  to  the  ray  itself, 
and  therefore  the  angle  made  by  this  projection  with  the  tiace  BD 
will  be  equal  to  the  angle  made  by  the  ray  with  the  vertical  plane 
But,  referring  to    PI.  IX.,    Fig.   80,   we  see  that  in   the  triangle 


82  sriADixG. 

LL'L,,  containing  lli  j  angle  LLiL'  made  by  the  ray  LL,  with  tlie  ver- 
tical plane,  tlie  side  L'L,  is  the  hypotliennse  of  tlic  triangle  L'i  li,, 
each  of  A\hose  other  sides  is  equal  to  LL'.  Hence  in  PI.  IX., 
Fig.  96,  take  any  distance,  Be,  make  AB  peri)endicular  to  BD',  arid 
on  it  lay  off  BDrrBc,  tlien  make  BD'i^Dc,  join  D  and  D',  and  DD' 
will  be  tlie  projection  of  a  ray  upon  the  plane  BD',  and  BD'D  will 
be  the  true  size  of  the  ajigle  made  by  the  light  with  tlie  vertical 
plane;  it  being  understood  that  tlie  plane  BD',  though  in  space 
perpendiculai-  to  tlic  vertical  iilaue,  is,  in  the  figure,  rejn'esented  as 
revolved  over  towards  the  right  till  it  coincides  with  the  vertical 
plane  of  projection,  and  with  the  paper. 

191.  We  are  now  ready  to  find  points  in  the  curve  of  shade,  oo' 
is  the  verticnl  projection  of  a  small  circle  parallel  to  the  plane  BD', 
and  aLso  of  its  tangent  rays.  The  circle  og'o\  on  oo'  as  a  diameter, 
represents  the  same  circle  revolved  about  oo'  as  an  axis  and  into 
the  vertical  plane  of  projection.  Drawing  a  tangent  to  og'o' ^ 
l)arallul  to  DD',  we  find  g\  a  visible  point  of  the  curve  of  shade, 
whicli,  when  the  circle  revolves  back  to  the  position  oo' ^  appears  at 
J/,  since,  as  the  axis  oo'  is  in  the  vertical  plane,  an  arc,  g'g^  described 
about  that  axis,  must  be  ye?*^«ea//y  projected  as  a  straight  line.  (See 
An.  yi.) 

In  a  precisely  similar  manner  ai"e  found  h.^  Jc,  w?,  and  f.  At  A 
and  B,  rays  are  also  evidently  tangent  to  the  sphere.  Through 
A,  y,  «S:c.,  to  B  the  visible  i)ortiou  of  the  curve  of  shade  may  now 
be  sketch''d. 

192.  The  niost  highly  illuminated  point  is  90°  distant  fioin  tli< 
grcai  circle  of  shade;  Tience,  on  ABQ,  the  revolved  position  of  a 
great  circle  which  is  perpendicular  to  the  circle  of  shade,  lay  oft* 
]c'Ql=^qV>  =  \\ic  chord  of  90°,  and  revolve  this  ]jerpendicular  circle 
back  to  the  })osi»ion  qq',  when  Q  will  be  found  ^t,  r'.  But  th«? 
brilliant  point,  as  it  appears  to  the  eye,  is  not  the  one  which  receivei 
most  light,  but  the  one  that  reflects  most,  and  this  point  is  midway, 
iu  space,  between  r'  and  r,  i.e.  at  P,  found  by  bisecting  QB,  and 
drawing  RP;  for  at  K  the  incident  ray  whose  revolved  position  is 
parallel  to  Qr  or  DD',  and  the  reflected  ray  whose  revolved  posi- 
tion is  parallel  to  rB,  make  equal  angles  with  R'R,  the  perpen- 
dicular (normal)  to  the  surface  of  the  sphere.    See  PL  X.,  Fig.  D. 

193.  In  regard  now  to  the  second  general  division  of  the  problem 
— the  distribution  of  tints  ;  a  small  oval  space  around  P  should  be 
It  It  blank.  The  first  stripe  of  dark  tint  reaches  from  A  to  B,  along 
tlie  curve  of  shade,  and  the  successive  stripes  of  the  same  tint  extend 
to  B<7A  on  one  side  of  BAvV,  and  to  octjo  on  the  other.     Then  take 


SHADING.  83 

ca  lighter  tint  on  the  lower  half  of  the  next  zone,  and  a  still 
lighter  one  on  its  upper  half  (2)  and  (3).  In  shading  the  next 
zone,  use  an  intermediate  tint  (3-4),  and  in  the  zone  next  to  P  a 
very  light  tint  on  the  lower  side  (4),  and  the  lightest  of  all  on 
the-  upper  side  (5).  After  laying  on  these  preliminary  tints, 
even  up  all  sudden  transitions  and  minor  irregularities  as  in 
other  cases, 

194.  Ex.  5°.  Shades  and  Shadows  on  a  Model.  PL 
XL,  Figs.  1  and  2.  General  Description. — This  plate  contains 
two  elevations  of  an  architectural  Model.  It  is  introduced  as 
affording  excellent  practice  in  tinting  and  shading  large  surfaces, 
and  useful  elementary  studies  of  shadows.  The  construction  of 
these  elevations  from  given  measurements  is  so  simple,  that  only 
the  base  and  several  centre  lines  need  be  pointed  out. 

QR  is  the  ground  line.     ST  is.a  centre  line  for  the  flat  topped 
tower  in  Fig.  1.     UVis  a  centre  line  for  the  whole  of  Fig.  2,  ex- 
cept the  left-hand  tower  and  its  pedestal.     WX  is  a  centre  line 
for  the  tower  through  which  it  passes.     YZ  is  the  centre  line  for 
the  roofed  tower  in  Fig.  1.     The  measurements  are  recorded  in 
full,  referred  to  the  centre  lines,  base  line,  and  bases  of  the  towers, 
\vhich  are  the  parts  to  be  first  drawn. 
Graphical  construction  of  the  shadows. 

1".  The  roof,  D — D'D",  casts  a  shadow  on  its  tower.  The  point, 
EE',  casts  a  shadow  where  the  ray,  Ee,  pierces  the  side  of  ^le  tower. 
e  is  one  projection  of  this  point ;  e',  the  other  projection  of  the  same 
point,  is  at  the  intei'section  of  the  line  ee"e'  with  the  other  projection, 
E'd',  of  the  ray.  Tlie  shadow  of  a  line  on  a  parallel  plane  (162)  is 
parallel  to  itself,  hence  e'f\  parallel  to  E'F',  is  the  shadow  of  E'F'. 

The  shadow  of  DE — D'E'  joins  e'  with  the  shadow  of  D — ^D'. 
The  point  c?,  determined  by  the  ray  Dc?,  is  one  projection  of  the 
latter  shadow  ;  the  other  projection,  cl\  is  at  the  intersection  of 
dcl'd'  witli  tlic  other  projection,  T>'cl\  of  the  ray,  d'  is  on  the  side 
of  the  tower,  produced,  hence  e'd'  is  only  a  real  shadow  line  from 
e  till  it  intersects  the  edge  of  the  tower. 

Remembering  that  the  direction  of  the  light  is  supposed  to 
change  with  each  position  of  the  observer,  so  that  as  he  faces  each 
side  of  the  model,  in  succession,  the  light  comes  from  left  to  right 
and  from  behind  his  left  shoulder,  it  appears  that  the  point,  DD", 
casts  a  shadow  on  the  fece  of  the  tower,  seen  in  Fig.  2,  and  that 
\y"d""  will  be  the  position  of  the  ray,  through  this  point,  on  F'ig, 


84"  SHADIXG. 

1.  The  point  tV"  is  therefore  one  projection  of  the  shadow  of 
I)D".  The  other  is  at  d"'\  the  intersection  of  the  lines  d"'d""  with 
Y)d"'\  the  other  projection  of  the  ray.  Likewise  EE"  casts  a 
shadow,  e"'e"'\  on  the  same  face  of  tlie  tower,  produced.  DD'", 
being  parallel  to  the  face  of  the  tower  now  being  considered,  its 
shadow,  d""q^  is  parallel  to  it.  The  line  from  d""  towards  e"", 
till  the  edge  of  the  tower,  is  the  real  i)ortiou  of  the  shadow  of 
])K_D"E". 

From  the  foregoing  it  will   be  seen  how  most  r)f  these  sha- 
dows are  found,  so  that  each  step  in  the  process  of  finding  similar 
hadows  will  not  be  repeated. 

2°.  The  body  of  the  building — or  model — casts  a  shadow  on  the 
roofed  tower,  beginning  at  AA'  (lOl).  The  shadow  of  BB'  on  tlie 
side  of  this  tower  is  bb\  ibund  as  in  previous  cases,  and  A'b'  is  the 
shadow  of  AB — A'B'.  From  b'  downwards,  a  vertical  line  is  the 
shadow  of  the  verti(;al  corner  edge  of  tlie  body  of  the  model  upon 
the  parallel  face  of  the  tower. 

3'.  The  line  CC" — C,  which  is  perpendicular  to  the  side  of  the 
roofed  tower,  casts  a  shadow,  C'c',  in  the  direction  of  the  projection 
of  a  ray  of  light  on  the  side  of  the  tower. 

4°.  In  Fig.  1,  a  similar  shadow,  »Y,  is  cast  by  the  edge  s' — sa' 
or"  the  smaller  pedestal. 

5°.  In  Fig.  2,  is  visible  the  curved  shadow,  c''rg^  cast  by  the 
vertical  edge,  at  c",  of  the  tower,  on  the  curved  part  of  tlie  pedestal 
of  the  to\i'er.  The  point  r/  is  found  by  drawing  a  ray,  G'C — G^, 
which  meets  the  upper  edge  of  the  pedestal  at  r/C.  The  point  c'l 
the  intersection  of  the  edge  of  the  tower  with  the  curved  part  of 
the  pedestal,  is  another  point.  Any  intermediate  point,  as  r,  is 
found  by  drawing  the  ray  R'r'/  r'  is  then  one  projection  of  the 
sh.idow  of  li'K,  and  the  other  is  at  the  intersection  of  the  line  r'r 
with  the  other  projection  llrof  the  ray.  These  are  all  the  shadows 
which  are  very  near  to  the  objects  casting  them. 

0".  The  flat  topped  tower  casts  a  shadow  on  the  roof  of  the  body. 
The  upper  back  corner,  IIII',  casts  a  shadow  on  the  roof,  of  which 
h  is  one  ])rojection  aii.l  ''  the  other.  The  back  upper  edge  II — IIT' 
oeing  parallel  to  the  roof,  the  short  shadow /i'A",  leaving  the  roof  at 
A",  is  pai-allel  to  HI'.  The  left  hand  back  edge  IIJ — IIM'  casts  a 
Bha<low  on  the  roof,  of  which  hh'  is  one  point.  The  point  .IJ'  casta 
a  shadow,;}"  on  the  roof  produced,  h'j  is  therefore  a  real  shadow 
only  till  it  leaves  the  actual  loof  at  u. 

V.  The  same  tower  casts  a  shadow  on  the  vertical  side  of  the 
body,  of  which  j"j"\  found  as  in  previous  cases,  and  i«,  are  pointa 


PL-'X. 


/H' 


'i  / 


/■'  I    :;•     \\y'h    >V'    ^M      \ 


^<r. 


B. 


D 


SHADING.  85 

the  upper  back  point,  KK',  of  the  shaft  of  the  tower,  casts  a  sha- 
dow, kh',  which  is  joined  with  /'",  giving  the  shadow  of 
JK — J'K'.  From  h' ,  k'l  io  the  vertical  shadow  line  of  the  left- 
liand  back  edge  of  the  fiat-topped  tower  on  the  parallel  plane 
of  the  side  of  the  body  of  the  model. 

8°.  The  same  tov/er  casts  a  shadow  on  the  curved — cylindrical 
part  of  the  pedestal.  To  find  the  point  m' ,  of  shadow,  draw  a 
ray,  MC,  Fig.  2,  intersecting  the  upper  edge  of  the  pedestal  at 
C,  which  is  therefore  one  projection  of  the  shadow  of  the  point 
MM'.  The  other  projection,  m' ,  of  the  same  shadow,  is  at  the 
intersection  of  the  other  projection,  Wm' ,  of  the  ray,  with  the 
other  projection,  m'Qi' ,  of  the  edge  of  the  pedestal.  The  point 
of  shadow,  nn' ,  cast  by  the  point  XX',  is  similarly  found,  and  so 
is  the  point  oo' ,  cast  by  the  point  00'  of  the  front  right-hand 
edge  of  the  tower.  Make  m'm"  =  n'o' ,  and  find  intermediate 
points,  v'v",  as  rr'  was  found,  and  the  curved  shadow  on  the 
cylindrical  part  of  the  pedestal  will  then  be  found. 

9".  From  n'  and  o',  vertical  lines  are  the  shadows  of  opposite 
diagonal  edges  of  the  tower,  on  the  vertical  face  of  the  main 
pedestal. 

li)\  This  flat-topped  tower  also  casts  a  shadow  on  the  side  of 
the  roofed  tower.  The  right  back  corner,  H — I',  of  the  top,  casts 
the  shadow  h"'h""  on  tlie  side  of  the  roofed  tower,  through 
which  the  shadow  line,  h""x  is  drawn,  jDarallel  to  the  line 
H — H'l'  which  casts  it.  The  right-iiand  top  Ime,  I' — IH,  being 
perpendicular  to  the  plane  of  the  sides  of  this  tower,  casts  the 
shadow  li""i'  upon  it,  parallel  to  the  projection  of  a  ray  of  light. 
(162.)  This  shadow  line  is  real,  only  till  it  leaves  the  tower  at 
z  ; — i'  being  in  the  plane  of  the  side  of  the  tower  produced — and  it 
completes  all  the  shadows  visible  in  the  two  elevations. 

Errors  in  Shading,  Relative  darkness  of  the  light  and  shade,  etc. 

195.  The  most  frequent  faults  to  guard  against  are — 1st.  A 
Irush  too  wet,  or  too  long  applied  to  one  part  of  the  figure  j  giving 
a  ragged  or  spotty  appearance. 

2d.  Outlities  inked  in  Hack,  whereas  the  form  and  outlines  of 
actual  objects  are  indicated,  not  by  black  edges,  but  by  contrast 
of  shade  only,  with  edges  lighter  than  other  parts. 

3^.  Much  too  little  contrast  between  the  shading  of  the  parts  in 


86  SHADING. 

the  light  and  those  iu  the  dark.  The  former  should  generally, 
as  on  a  cylinder,  be  very  much  lighter,  or  not  one  quarter  as 
dark  as  the  parts  in  the  dark.  This  is  confirmed  by  examining 
photographs  of  objects  illuminated  in  the  manner  supposed  in 
this  chapter. 

19G.  There  are  Jive  methods  of  shading,  as  follows  : 

1st.  Softened  loet  shading.  This  is  done  with  a  large  brush, 
quite  wet,  and  is  applicable  to  large  figures. 

2d.   Softened  dry  shading.     This  is  done  Avith  a  brusli,  nearly 
dry,  and  is  applicable  to  figures  of  the  size  of  those  on  PI.  X., 
or  not  much  larger. 

od.  Sltading  by  superposed  flat  tints.  This  method  is  very  neat 
and  effective  for  large  figures,  not  to  be  closely  examined.  It  con- 
sists of  the  preliminary  stripes  of  the  "Zd  method  evenly  and 
smoothly  done,  w^YZso?/^  softening  their  edges  (183  d). 

Uh.   Stippling,  or  dot  shading.     This  is  done  with  a  fine  pen 
as  in  making  sand  in  copographical  drawing. 

bth.  Line  shading.  This  is  done  by  means  of  lines  of  graded 
size  and  distance  apart ;  as  in  wood  and  lithographic  mechanical 
engravings. 

The  details  of  these  methods  are  further  exjjlained  in  my 
"Drafting  Instruments  and  Operations." 


PL.XI. 


c 


DlVISIOISr    FOURTH. 

ISOMETRIC  A  L  AND  OBLIQUE  PROJECTIONS. 


CHAPTER  I. 

FIRST   PRINCIPLES   OF   ISOMETRICAL   DRAWING. 

197.  It  is  the  object  of  this  Division  to  explain  some  methods 
of  making  drawings,  especially  of  details  and  various  small  work, 
which  combine  the  intelligibleness  of  pictorial  figures,  with  the 
exactness  of  common  projections  (Div.  I.).  Such  drawings  pos- 
sess, among  others,  the  advantage  of  being  more  readily  under- 
stood by  workmen  unacquainted  with  ordinary  projections,  than 
plans  and  elevations  might  be. 

We  shall  therefore  now  explain  the  methods  called  Isometri- 
cal  and  Oblique  Projections,  taking  up  the  former  first. 

198.  Isometrical 2)rojection  is  that  in  which  a  solid  right  angle, 
like  that  at  the  corner  of  a  cube,  is  placed  so  that  the  three  plane 
right  angles  which  bound  it  appear  equal  in  the  projection. 

The  elementary  princijDles  of  this  projection  are  most  simply 
explained,  as  follows,  by  reference  to  a  cube,  since  this  is  the 
simplest  possible  rectangular  body. 

For  clearness,  we  have  to  distinguish  in  a  cube,  its  edges,  its 
face  diagonals,  and  its  body  diagonals. 

199.  Accordingly,  let  PI.  XII.,  Fig.  98,  represent  a  cube  whose 
front  face  is  parallel  to  V.  This  face  will  therefore  be  the  only 
one  visible  in  elevation,  and  will  appear  in  its  real  size. 

Next  suppose  the  cube  to  be  turned  horizontally  45°,  when 
two  of  its  vertical  faces  will  be  shown  equally,  as  in  Fig.  99, 
though  not  in  their  real  size. 

Finally,  suppose  the  cube  to  be  turned  up  at  its  back  corner 
dd',  about  any  axis  as  ef,  parallel  to  the  ground  line,  until  the 


88  FIRST    PRIXCIPLES   OF   ISOMETRICAL   DRAWIXG. 

top  and  the  two  faces  shown  in  Fig.  99  all  appear  equal,  as  in 
Fig.  100.  This  evidently  can  be  done,  for  in  Fig.  99  the  top 
face  is  not  seen  at  all  in  vertical  projection;  but  invert  the  figure, 
taking  plan  for  elevation,  and  it  is  fully  seen,  and  the  vertical 
faces  are  unseen;  hence  there  must  be  one  intermediate  position, 
where,  as  in  Fig.  100,  the  three  faces  seen  in  Fig.  99  will  be 
seen  equally. 

iiOO.  Results. — 1°.  The  equality  of  the  projections  of  the 
three  \\s,\\Aq  faces  of  the  cube,  when  inclined  as  in  Fig.  100,  in- 
cludes the  equality  of  the  projections  of  its  edges.  Hence  (Div. 
I.,  Art.  11),  these  edges  also  being  equal  in  space,  are  equally 
inclined  to  the  plane  of  projection. 

2°.  The  equal  projections  of  the  faces  include  the  equal  pro- 
jections of  their  like  angles.  Hence  the  three  angles  at  C,  and 
tlie  angles  equal  to  them  at  D,  o,  p,  are  angles  of  120°  each. 
The  remaining  angles  of  the  projections  of  the  faces  are  of  60° 
each. 

r.  Lines  like  Ce",  CD,  C"f",  Fig.  100,  equal,  and  equally 
inclined  to  a  plane,  and  jiroceeding'  from  one  point,  C,  must 
terminate  in  a  plane  parallel  to  the  given  plane.  Hence  the 
three  face  diagonals  e"f",  f"n,  ne',  are  all  parallel  to  the  plane 
of  projection,  and  therefore  appear  in  their  real  size.  Hence  in 
(199)  we  might  have  said,  incline  the  cube,  until  three  visible 
face  diagonals,  h'd'  —  ^X);  a"h'—K.'V>;  «"(/'  — A'D,  Fig.  A, 
become  parallel  to  the  plane  of  i^rojection,  XY.  All  other  lines 
of  Fig.  100  appear  less  than  their  real  size. 

4°.  By  3°,  e"f"n  is  evidently  an  equilateral  triangle  ;  and,  by 
1°  and  2°,  all  the  other  lines  make  equal  angles  of  30''  with 
these.  Hence  the  perimeter  Df'onpe"  is  a  regular  hexagon, 
and  the  projections  of  the  visible  foremost,  and  invisible  hind- 
most corners  of  the  cube  coincide  at  C.  Hence  the  diagonal  of 
the  cube,  which  joins  these  points,  is  perpendicular  to  the  plane 
of  projection. 

201.  Definitions.  Fig.  100:  C  is  the  isometric  centre,  Ce", 
Qf",  C'n,  are  the  isometric  axes.  Lines  parallel  to  these  axes 
are  isometric  lines  ;  others  are  non-isometric  lines.  Planes  par- 
allel to  the  faces  of  the  cube  arc  isometric  planes. 

The  edges  of  a  cube,  or  similar  rectangular  body,  are  the  lines 


FIRST   PRINCIPLES   OF   ISOMETRICAL   DRAWING.  89 

whose  dimensions  would  naturally  be  desired.  Hence  it  is  usual 
to  make  Ce",  Cf,  He",  etc..  Fig.  100,  equal  to  the  edges  of  the 
cube  itself ;  as  in  Fig.  101,  Such  a  figure  is,  for  distinction, 
called  the  Isometrical  Drawing  of  the  cube,  and  it  is  the  isomet- 
rical  projection  of  an  imaginary  larger  cube,  which  is  to  Fig.  101 
as  Fig.  99  is  to  Fig.  100. 

202.  Another  demonstration*  The  first  principles  of  iso- 
metrical projection  may  be  developed  from  the  following 

Proposition.  The  plane  luhich  is  perpendicular  to  the  body 
diagonal  of  a  cube,  is  equally  inclined  to  its  faces  and  edges. 

Let  abcd—a'b'c'd'a"b"c"d",  PI.  XXL,  Fig.  A,  be  the  plan 
and  elevation  of  a  cube,  shown  as  in  PI.  XII.,  Fig.  99,  only 
that  the  ground  line,  GL,  is  inclined,  to  permit  the  construction 
of  the  isometrical  figure  in  an  upright  position,  as  in  Fig. 
100,  by  direct  projection  from  the  given  elevation. 

The  body  diagonal,  ac — a'c" ,  parallel  to  V,  is  the  common 
hypothenuse  of  three  equal  right-angled  triangles,  similarly  situ- 
ated relative  to  the  cube.  One  base  of  each  of  these  triangles  is 
an  edge,  beginning  at  cc"  ;  the  other  is  a  face  diagonal,  begin- 
ning at  aa' . 

Thus,  one  of  these  three  triangles  is  ac  —  a'c'c" ,  which,  being 
parallel  to  V,  shows  its  real  size  on  V.  Another  has  for  its  bases 
the  edge  be — b"c"  and  the  face  diagonal  ab — a'b" ;  and  the 
third  has  for  its  bases  the  edge  dc — d"c" ,  and  the  face  diagonal 
ad~a'd". 

!N^ow  since  these  triangles  arc  tbus  equal,  and  similarly  i)laced 
on  the  cube,  their  common  hypothenuse,  the  body  diagonal 
a'c" ,  making  equal  angles  with  the  three  face  diagonals  which 
meet  at  aa' ,  also  makes  equal  angles  with  the  faces  containing 
these  diagonals. 

Hence  a  plane  of  projection  XY,  or  Ti,  perpendicular  to  this 
body  diagonal,  is  equally  inclined  to  the  three  faces  of  the  cube 
"which  meet  at  aa' ,  or  at  cc" ;  which  agrees  with  the  enunciation. 

From  this  conclusion  follow  all  the  other  particulars  in  the 
preceding  articles. 

In  making  the  figure,  H  and  the  plane  XY  (Vi)  are  both  per- 
pendicular to  Y.     Hence  Cn=cc" ;  'Do=dd"j  etc. 

*  This  may  be  omitted  at  discretion,  but  may  be  preferred  by  Teachers 
and  others,  as  fresher,  and  more  concise  and  strictly  geometrical. 


CHAPTER  II. 

PROBLEMS   INVOLVIXG    OKLY   ISOMETRIC    LINES. 

203.  Prob.  1.  To  construct  the  isomctrical  jJrojections,  and 
draioings  of  cubes,  and  a  rectangular  hloch  cut  from  a  corner  of 
one  of  them. 

The  principles  of  the  last  chapter  yield  several  simple  con- 
structions, as  follows: 

204.  First  Method.  Draw  an  equilateral  triangle,  as  e"f"n. 
Fig.  100,  each  of  whose  sides  shall  be  equal  to  a  face  diagonal  of  the 
cube.  Then  (200,  4°)  draw  two  lines,  as  e"D  and  e"Q,  with  the 
30°  angle  of  the  30°  and  60°  triangle,  making  angles  of  30°  with 
each  side  of  the  triangle  e"f"n,  at  each  extremity.  These  lines 
will  intersect  as  at  C,  D,  o,  etc.,  forming  the  isometric  j9ro/ec^«ow 
of  the  cube.  By  extending  Q,e" ,  C/",  Qn,  till  equal  to  the  edge 
of  the  cube,  and  joining  their  new  outer  extremities,  the  projec- 
tion will  be  converted  into  the  isometric  drawing. 

205.  Second  3Iethod.  Fig.  100.  Draw  the  isometric  axes  indefi- 
nitely, Ce"  and  Cf"  each  making  angles  of  30°  with  a  horizontal 
EF,  on  which  lay  off  half  of  a  face  diagonal  e/.  Fig.  99,  each  way 
from  C,  giving  E  and  F.  Then  perpendiculars  at  E  and  F  will 
limit  the  right  and  left  axes  at  e"  and  f".  Cn  is  then  made 
equal  to  Ce"  or  C/",  and  the  remaining  edges  are  drawn  parallel 
to  these  three.  From  this  ^Jrojeciion,  make  the  draiviyig  as  in 
the  first  method. 

206.  Third  Method.  Drawing  tlie  axes  as  before,  lay  off  on  each 
the  true  length  of  an  edge  of  the  cube,  and  complete  the  figure 
as  just  described.     This  at  once  makes  the  draining. 

207.  Fourth  Method.  Fig.  101.  With  centre  C  and  radius  equal 
to  an  edge  of  the  cube,  draw  a  circle,  and  in  it  inscribe  a  regu- 
lar hexagon  in  the  position  shown,  adding  the  alternate  radii 
Ca,  Ch,  Cc,  which  again  gives  the  draiving  of  the  cube. 

Let  a  prismatic  block  be  cut  from  the  front  corner  of  this 
cube.    Suppose  the  edge  of  the  cube  to  be  five  inches  long,  drawn 


PROBLEMS  rSTVOLVING    OXLY   ISOMETRIC   LINES.  91 

to  a  scale  of  one-fifth.  Let  Ca'=2  inches;  CJ'^S  inches;  Cc'  = 
1  inch.  Lay  off  these  distances  upon  the  axes,  and  at  a',  V ,  c' , 
draw  isometric  lines  which  will  be  the  remaining  visible  edges  of 
the  block. 

Examples. — \8t.  Draw  a  square  panel  in  each  visible  face. 

'^d.  Draw  a  square  tablet  in  each  visible  face. 

od.  Let  a  cube  l^-  inches  on  each  edge  be  cut  from  every  visi- 
ble corner  of  the  cube. 

208.  Prob.  2.  To  find  the  shade  lines  on  a  cube,  and  the 
shadows  of  isometric  lines  upon  isometric  pkmes. 

Three  faces  of  the  cube,  PL  XII.,  Fig.  102,  will  be 
illuminated  by  light  passing  in  the  direction  LL',  from  a  to  p. 
Two  of  these  faces,  ao7iC,  and  aCqD  are  visible.  The  under  and 
right-hand  faces,  ojipC,  npqC,  and  pqDC,  are  in  the  dark,  hence 
by  the  rule  (18)  the  edges  on,  nC,  Cq  and  qD  are  to  be  inked 
heavy. 

The  shadoivs  of  al)=-oa  produced  ;  and  of  ad=J)a  produced. 
ai  is  perpendicular  to  aCqD,  hence  (162,  Ath)  its  shadow  will  be 
in  the  direction  of  the  projection  of  the  light  upon  that  face. 
But  aq  is  evidently  the  projection  of  the  ra}',  LL'  upon  aCqD', 
and  as  a  is  where  ab  meets  aCqJ)  (1G2,  btJi),  ah'  is  the  shadow  of 
ab  on  aQqT). 

In  like  manner,  ad'  is  the  shadow  of  ad  upon  aonO. 

Otherwise.  A  plane  containing  oa  and  ap,  will  contain  rays 
through  points  of  oab,  and  will  also  contain  jog  and  W'ill  therefore 
cut  aQqJ)  in  the  line  aq.  Hence  rays  from  all  points  of  ab  will 
pierce  aCqJ}  in  aq.  Hence  ab'  is  the  shadow  of  ab.  Again  ; 
Idap  is  the  plane  of  rays,  containing  DcZ  and  pn,  and  cutting  the 
face  aonQ  in  the  line  an.  Hence  d'  is  the  shadow  of  d,  and  ad', 
that  of  ad. 

Similar  constructions  apply  to  the  isometric  projections  and 
drawings  of  all  rectangular  bodies. 

Examples. — 1st.  Find  the  shadow  of  the  cube  on  the  plane 
of  its  lower  base. 

2t?.  Find  the  shadow  of  aL  (considered  as  Ca  produced)  upon 
the  rear  face  Dao. 

Zd.  Find  the  shadow  of  a  line  parallel  in  space  to  Ca,  upon 
the  face  Gaon, 


92  PROBLEMS   IXVOLTIXG    OXLT    ISOMETEIC    LIN"ES. 

Wi.  Find  the  shadow  of  Dq  produced,  on  tlie  plane  of  the 
lower  base  of  the  cube. 

209.  Pkob.  3.  To  construct  the  isometrical  drawing  of  a  car- 
penter's oil-stone  lox.  PL  XIIL,  Fig.  103.  This  problem  involves 
the  finding  of  points  which  are  in  the  planes  of  the  isometric 
axes. 

Let  the  box  containing  the  stone  be  10  inches  long,  4  inches 
wide,  and  3  inches  high,  and  let  it  be  drawn  on  a  scale  of  \. 

Assume  C  then,  and  make  Ca=4:  inches,  C5=10  inches,  and 
Cc=3^  inches,  and  by  other  lines  «D,  Db,  &c.,  equal  and  parallel 
to  these,  complete  the  outline  of  the  box. 

Eepresent  the  joint  between  the  cover  and  the  box  as  being  1 
inch  below  the  top,  aCh.  Do  this  by  making  Cj)  =  l  inch,  and 
through  p  drawing  the  isometric  lines  which  rejiresent  the 
joint. 

Sujipose,  now,  a  piece  of  ivory  5  inches  long  and  1  inch  wide 
to  be  inlaid  in  the  longer  side  of  the  box.  Bisect  the  lower  edge 
at  (I,  make  de=l  inch,  and  ee=l  inch.  Through  e  and  e',  draw 
the  top  and  bottom  lines  of  the  ivory,  making  them  2^  inches 
long  on  each  side  of  de',  as  at  e't,  e't'.  Draw  the  vertical  lines 
at  t  and  i',  which  will  complete  the  ivory. 

210.  To  show  another  way  of  representing  a  similar  inlaid 
piece,  let  us  suppose  one  to  be  in  tlie  top  of  the  cover,  5  inches 
long  and  1\  inches  wide.  Draw  the  diagonals  ab  and  CD,  and 
through  their  intersection,  o,  draw  isometric  lines ;  lay  off  oo'= 
2^  inches,  and  cio"  =  |  of  an  inch,  and  lay  off  equal  distances  in 
the  opposite  directions  on  these  centre  lines. 

Through  o',  o",  &c.,  draw  isometric  lines  to  complete  the 
representation  of  the  inlaid  jiiece. 

PL  XII.,  Fig,  B,  suggests  other  exercises,  and  illustrates  the 
gaining  of  room  by  making  the  longest  lines  of  the  figure  hori- 
zontal, as  is  often  done  in  practice,  when  the  longest  lines  of  the 
original  object  are  horizontal,  since  it  is  not  tlie  position,  but 
the /orm  of  the  figure  wliicli  makes  it  isometrical. 

Examples. — 1st.  Reconstruct,  isometricalh',  two  courses  of 
any  of  the  examples  in  ])rick-W()rk  on  PL  V. 

'2d.  Do  likewise  witli  any  of  the  figures  (4G-49)  of  PL  VII., 
assuming  any  convenient  width  for  the  timbers  in  each  case. 


PL.Xll. 


r 


r 


PROBLEMS   INVOLVING    ONLY    ISOMETRIC    LINES.  93 

211.  Prob.  4.  To  represent  the  same  hox  {Frob.  3)  tvith  the 
cover  reiiiovcd.     PL  XIIL,  Fig.  104. 

This  j^roblem  involves  the  finding  of  the  positions  of  jooints 
not  in  the  given  isometric  planes. 

Suijposing  the  edges  of  the  box  to  be  indicated  by  the  same  let- 
ters as  are  seen  in  Fig.  103,  and  suj)posing  the  body  of  the  box  to 
be  drawn,  10  inches  long,  4  inches  wide,  and  2i}  inches  high;  then, 
to  find  the  nearest  upper  corner  of  the  oil  stone,  lay  oif  on  Cb  li- 
inches,  and  through  the  point  /,  thus  found,  draw  a  line,  ff, 
parallel  to  Ca.  Oiiff,  lay  oif  1  inch  at  each  end,  and  from  the 
points  Jih',  thus  found,  erect  perpendiculars,  as  hn,  each  f  of  an 
inch  long.  Make  the  further  end,  x,  of  the  oil  stone  li- 
inches  from  the  further  end  of  the  box,  and  then  complete  the 
oil  stone  as  shown.  To  find  the  panel  in  the  side  of  the  box,  lay 
off  2  inches  from  each  end  of  the  box,  on  its  lower  edge;  at  the 
points  thus  found,  erect  perpendiculars,  of  half  an  inch  in  length, 
to  the  lower  corners,  as  p,  of  the  panel ;  make  the  2:>anels  1^ 
inches  wide,  as  at  ^t/;',  and  -^  an  inch  deep,  as  atpr,  Avhich  last 
line  is  parallel  to  Cd. ;  and,  Avith  the  isometric  lines  through 
r,  p,  p' ,  and  ;*,  completes  the  panel. 

212.  The  manner  of  shadimj  tliis  figure  Avill  now  be  explained. 
The  top  of  the  box  and  stone  is  lightest.     Their  euds  are  a 

trifle  darker,  since  they  receive  less  of  the  light  which  is  diffused 
through  the  atmosphere.  The  shadoAv  of  the  oil  stone  on  the 
top  of  the  box  is  much  darker  than  the  surfaces  just  mentioned. 
The  shadow  of  the  foremost  vertical  edge  of  the  stone  is  found 
in  precisely  the  same  Avay  as  was  the  shadow  of  the  Avire  iTpon  the 
top  of  the  cube,  PI.  XII.,  Fig.  102.  The  sides  of  the  oil  stone 
and  of  the  box,  Avhich  are  in  the  dark,  are  a  little  darker  than  the 
shadoAv,  and  all  the  surfaces  of  the  panel  are  of  equal  darkness 
and  a  trifle  darker  than  the  other  dark  surfaces.  In  order  to 
distinguish  the  separate  faces  of  the  panel,  Avhen  they  are  of  the 
same  darkness,  leave  their  edges  very  light.  The  little  light 
Avliich  those  edges  receive  is  mostly  perpendicular  to  them,  re- 
garding them  as  rounded  and  polished  by  use.  These  light  lines 
are  left  by  tinting  each  surface  of  the  panel,  separately,  with  a 
small  brush,  leaving  the  blank  edges,  Avhich  may,  if  necessary, 
be  afterAvards  made  jierfectly  straight  hx  inking  them  Avith  a 


di  PKOBLEMS    IXVOLVI^'G    OXLT    ISOMETRIC    LINES. 

light  tint.  The  upper  and  left-hand  edges  of  the  panel,  and  all 
the  lines  corresijonding  to  those  which  are  heavy  in  the  previous 
figures,  may  be  ruled  with  a  dark  tint.  In  the  absence  of  an 
engraved  copy,  the  figures  will  indicate  tolerably  the  relative 
darkness  of  the  different  surfaces ;  1  being  the  lightest,  and  the 
numbers  not  being  consecutive,  so  that  they  may  assist  in  denot- 
ing relative  differences  of  tint.  When  this  figure  is  thus  shaded, 
its  edges  should  not  be  inked  with  ruled  lines  in  black  ink,  but 
should  be  inked  with  pale  ink. 

Examples. — 1st.  Eeconstruct,  isometrically,  any  of  the  figures 
53,  54,  55,  58,  on  PL  VIL;  or  figures  63,  64,  65,  67,  of  PI.  VIII., 
omitting  the  bolts,  [The  few  non-isometric  lines  that  occur, 
being  in  given  isometric  planes,  can  be  located  by  a  litle  care.] 

2d.  Construct  the  isometrical  drawing  of  a  low  stout  square- 
legged  bench,  and  add  the  shadows. 

3c?.  Construct  the  isometrical  drawing  of  a  cattle  yard  -with 
pens  of  various  sizes  and  heights. 

4:th.  Construct  the  isometrical  drawing  of  a  shallow  partitioned 
drawer,  or  box,  with  the  shadoAvs. 

5th.  Make  the  isometrical  drawing  of  a  cellar  having  a  wall  of 
irregular  outline,  and  showing  the  chimneys,  partitions,  bins,  etc. 


CHAPTER  III. 

PnOBLEMS    IXVOLVING   NOX-ISOMETRICAL   LINKS. 

213.  SixcE  non-isometiical  lines  do  not  appear  in  their  true  size, 
each  point  in  any  of  them,  when  it  is  in  an  isometric  plane,  must  be 
ocated  by  two  isometric  lines,  which,  on  the  object  itself,  are  at 

light  angles  to  each  other.  Points,  not  in  any  known  isometrical 
plane  must  be  located  by  three  such  co-ordinates,  as  they  are  called, 
from  a  known  point. 

214.  Prob.  5.    To  construct  the  isometrical  draining  of  the 
scarfed  splice,  shoivn  at  PL  VIII.,  Fig.  66.    Let  the  scale  be  ^,  or 
three-fourths  of  an  inch  to  a  foot.  In  this  case,  PI.  XIII.,  Fig.  105,  it 
will  be  necessary  to  reconstruct  a  portion  of  the  elevation  to  the  new 
scale  (see  PI.  XIII.,  Fig.  106),  where  A;;  =  l|-  feet,  7jA"  =  6  inches, 
and  the  proportions  and  arrangement  of  parts  are  like  PL  VIII. , 
Fig.  OG.     In  Fig.  105,  draw  AD,  3  feet;  make  DB  and  AA'  each 
one  foot,  and  draAv  through  A'  and  B  isometric  lines  parallel  to  AD 
Join  A — B,  and  from  A  lay  off  distances  to  1,  2,  3,  4,  equal  to  the 
corresponding  distances  on  Fig.  106.    Also  lay  off,  on  BK,  the  same 
distances  from  B,  and  at  the  points  thus  found  on  the  edges  of  the 
timber,  draw  vertical  isometric  lines  equal  in  length  to  those  which 
locate  the  corners  of  the  key  in  Fig.  106.     Notice  that  opposite 
sides  of  the  keys  are  parallel,  and  that  AV,  and  its  jiarallel  at  B, 
are  both  parallel  to  those  sides  of  the  keys  which  are  in  space  per- 
pendicular to  AB.     To  represent  the  obtuse  end  of  the  upjjcr  tim 
berof  the  splice,  bisect  AA',  and  make  va=Aa,  Fig.  106,  and  draw 
Aa  and  A'a.     Locate  m  by  va  produced,  as  at  ar,  Fig.  106,  and  a 
short  perpendicular=rm.  Fig.  106,  and  draw  mV  and  mV ;  W 
being  parallel  to  AA',  and  am  being  parallel  to  AV.     To  represent 
the  washer,  nut,  and  bolt,  draw  a  centre  line,  vv',  and  at  t,  the  mid- 
dle point  of  Vw,  draw  the  isometric  lines  tti  and  tie,  which  will  givi 
c,  the  centre  of  the  bolt  hole  or  of  the  bottom  of  the  washer.  A  point 
— coinciding  in  the  drawing  with  the  upper  front  comer  of  the  nut — 
is  the  centre  of  the  top  of  the  washer,  which  may  be  made  }  of  iui 
inch  thick. 


9G  rU015LKSIS    INVOLVIXG    NO.\-ISO-METKICAL    LINES. 


Tlnou<ili  the  above  point  diaw  isometric  lines,  rr'  and^>p',  and 
lay  oli' on  them,  Ironi  the  same  point,  the  radius  of  the  washer,  say 
2^  inches,  giving  four  points,  as  o,  through  Mhich,  if  an  isometric 
Bquare  be  drawn,  the  top  circle  of  the  washer  can  be  sketched  in  it^ 
being  tangent  to  the  sides  of  this  square  at  the  points,  as  o,  and 
elliptical  (-oval)  in  shape.  The  bottom  circle  of  the  washer  is  seen 
throughout  a  semi-circumference,  i.  e.  till  limited  by  vertical  tan- 
gents to  the  upper  curve. 

On  the  same  centre  lines,  ^ay  off  from  their  intersection,  the  half 
side  of  the  nut,  1  \  inches,  and  from  the  three  corners  which  will  be 
visible  when  the  nut  is  drawn,  lay  off  on  vertical  lines  its  thickness, 
1;!  inches,  giving  the  upper  corners,  of  which  c  is  one.  So  much 
being  done,  the  nut  is  easily  finished,  and  the  little  fragment  of  bolt 
projecting  through  it  can  be  sketched  in. 

Example. —  Construction  of  the  nut  when  set  ohliqudy.  The  not 
is  here  constructed  in  the  simplest  position,  i.  e.  with  its  sides  in  the 
direction  of  isometric  lines.  If  it  had  been  determined  to  construct 
it  in  any  oblique  position,  it  would  have  been  necessary  to  have 
constructed  a  portion  of  the  plan  of  the  timber  with  a  plan  of  the 
nut — then  to  have  circumscribed  the  plan  of  the  nut  by  a  square, 
parallel  to  the  sides  of  the  timber — then  to  have  located  the  cor- 
ners of  the  nut  in  the  sides  of  the  isometric  drawing  of  the  circum- 
scribed square.  Let  the  student  draw  the  other  nut  on  a  large  scale 
and  in  some  such  irregular  position.  See  Fig.  107,  Mhere  the  upper 
6gure  is  the  isometrical  drawing  of  a  squaie,  as  the  top  of  a  nut; 
this  nut  having  its  sides  oblique  to  the  edges  of  the  timber,  which 
are  supposed  to  be  parallel  to  ca. 

215.  Proi;.  G.  7b  make  an  isometrical  drawing  of  an  oblique 
timber  framed  into  a  horizontal  one.  PI.  XIII.,  Fig.108.  Let  the 
( riginal  be  a  model  in  which  the  ho'"izontal  piece  is  one  inch  square, 
and  lot  the  scale  be  \. 

Make  ag  and  aA,  each  one  inch,  complete  the  isometric  end  of  the 
horiziint.d  piece,  :ind  draw  ad^hn  ^wdi  gk.  Letf/J  — 2  inches,  ic  =  2J 
inches,  and  draw  bm.  Let  the  slant  of  the  oblique  timber  be  such 
that  if  «7=2  inches,  de  shnil  be  |  of  an  inch.  Then  by  (21.3)  de^ 
piirallel  to  ag,  will  give  ce,  whi(;h  docs  not  show  its  true  size.  Draw 
edges  parallel  lo  ce  through  b  and  m.  In  like  mannerycan  be  found, 
by  distances  taken  from  a  side  elevation,  like  PI.   VIII. ,  Fig.  59. 


21  (J.  Pro«.  7.     To  make  an  isometrical  drawing  of  a  pyramid 
ttandbaj    upon  a  recessed pcdcaUd.    PL  XIII.,  Fig.  109.    (From  a 


k 


PltOllLESIS    INVOLVING    NON-ISOAIETKICAL   LINKS.  97 

Model.)  Let  the  scale  be  ^.  Assuming  C,  construct  the  isometrio 
square,  CABD,  of  which  each  side  is  4i  inches.  From  each  of  its 
corners  lay  off  on  each  adjacent  side  1^  inches,  giving  points,  as  e 
and/'/  from  all  of  these  points  lay  off,  on  isometric  lines,  distances 
of  I  of  an  inch,  giving  points,  as  </,  /;,  and  h.  From  all  these  points 
now  found  in  the  upper  surface,  through  which  vertical  lines  can  be 
e^^jen,  draw  such  lines,  and  make  each  of  them  one  inch  in  length, 
and  join  their  lower  extremities  by  lines  parallel  to  the  edges  of  the 
lop  surface. 

Isometric  lines  through  <7,  m,  &c.,  give,  by  their  intersections,  the 
corners  of  the  base  of  the  pyramid,  that  being  in  accordance  with 
the  construction  of  this  model,  and  the  intersection  of  AD  and  CD 
is  o,  the  centre  of  that  base.  The  height  of  the  pyramid — ]f  inchea 
■ — is  laid  off  at  oa.  Join  v  with  the  corners  of  the  base,  and  the 
construction  will  be  complete. 

217.  To  find  the  shadows  on  this  model. — According  to  prin- 
ciples enunciated  in  Division  III.,  the  shadow  oi  fh  begins  at  h, 
and  will  be  limited  by  the  line  As,  s  being  the  shadow  of /*,  and  the 
intersection  of  the  ray/s  with  hs.  st  is  the  shadow  ofyF.  Accord- 
ing to  (208),  10  is  the  shadow  of  y,  and  joining  w  with  jo  and  q^  the 
opposite  corners  of  the  base,  gives  the  boundary  of  the  shadow  on 
the  pedestal,  ka  is  the  boundary  of  the  shadow  of  gk  on  the  lace 
ku.  The  heavy  lines  are  as  seen  in  the  figure.  If  this  drawing  is 
to  be  shaded,  the  numerals  will  indicate  the  darkness  of  the  tint  for 
the  several  surfaces — 1  being  the  lightest. 

218.  Prob.  8.  To  construct  the  isometrical  drawing  of  a  wall  in 

hatter^  with  counterforts,  and  the  shadoivs  on  theioall.  PI.  XIII., 
Fig.  110.  A  wall  in  batter  is  a  wall  whose  face  is  a  little  inclined 
to  a  vertical  plane  through  its  lower  edge,  or  through  any  horizon- 
tal line  in  its  face.  Counterforts,  or  buttresses,  are  projecting  parts 
attached  to  the  wall  in  order  to  strengthen  it.  Let  tho  scale  be 
one  fourth^  i.e.,  let  each  one  of  the  larger  spaces  on  the  scale 
marked  40,  on  the  ivory  scale,  bo  taken  as  an  inch. 

Assume  C,  and  make  CD=:2  inches  and  CI  =:  10  inches  in  the  righ 
and  left  hand  isometrical  directions.  Make  DF=6  inches,  EF=1^ 
inches,  FG=1  J  inches,  and  GH=1^  inches,  and  join  H  and  E,  all 
of  these  being  isometrical  lines.  Next  draw  EC,  then  as  the  top 
of  the  counterfort  is  one  inch,  vertically,  below  the  top  of  the  Avail, 
while  EC  is  not  a  vertical  line,  make  Fc?=-one  inch,  draw  de  paraU 
lei  to  FE  and  e/,  parallel  to  EH.  Make  ef  and  Ca  each  one  inch, 
at  /make  the  isometric  line/^=r|  of  an  inch,  and  at  a  make  ah  =  2 


98  PBOBLEMS    INTOLVIN'G    NON-ISOMKTKICAL   LINES. 

inches,  and  draw  hg.  Make  hc—\\  inches,  draw  ch  parallel  to  hg 
niul  complete  the  top  of  the  counterfort.  The  other  counto'fort 
is  similar  in  shape  and  similarly  situated,  i.  e.,  its  furthermost  lower 
corner,  A;,  is  one  inch  irom  I  on  the  line  IC,  while  HI  is  equal  and 
parallel  to  CE. 

219.  To  find  the  shadows  of  the  counterforts  on  the  face  of  Hu 
i?aU. — We  have  seen—  PI.  IX.,  Fig.  78 — that  when  a  line  is  per« 
pt-ndicular  to  a  vertical  plane,  its  shadow  on  that  plane  is  in  the 
direction  of  the  projection  of  the  iiglit  n\H)\)  the  same  plane,  and 
from  PI.  XII.,  Fig.  102,  that  the  i)rojection,  an,  of  a  ray  on  the  left 
hand  vertical  face  of  a  cube  makes  on  the  isometrical  drawuig  an 
angle  of  GO''  with  the  horizontal  line  nn'. 

Now  to  find  the  shadow  of  mn.  The  front  face  of  the  wall—  PI. 
XIII., Fig.llO— not beingverticaljdropapcrpendicular,  fo,  from  an 
upj)er  back  corner  of  one  of  the  counterforts,  upon  the  edge  o»a, 
produced,  of  its  base,  and  through  the  point  o,  thus  found,  draw  a 
line  parallel  to  CI.  nfop  is  then  a  vertical  jjlane,  pierced  by  inn, 
an  edge  of  the  further  counterfort  which  casts  a  shadow ;  and 
nj)  is  the  direction  of  the  shadow  of  m;i  on  this  vertical  plane. 
The  shadow  of  m7i  on  the  plane  of  the  lower  base  of  the  wall  is  of 
course  parallel  to  mn,  and  ^>  is  one  point  of  this  shadoAV,  hence  pq 
is  the  direction  of  this  shadow.  Now  n,  where  mn  pierces  the 
actual  face  of  the  wall,  is  one  point  of  its  shadow  on  that  face,  and 
q,  where  its  shadow  on  the  hoiizontal  i)Iaue  pierces  the  same  face, 
is  another  point,  hence  9tQ  is  the  general  direction  of  the  shadow 
of  tnn  on  the  front  of  the  wall,  and  the  actual  extent  of  this  shadow 
is  7U\  r  being  where  the  ray  ttir  pierces  the  front  of  the  wall. 

From  r,  the  real  shadow  is  cast  by  the  edge,  mu,  of  the  counter- 
fort, r,  the  shadow  of  m,  is  one  point  of  this  shadow,  and  s,  where 
um  produced,  meets  yn  i)roduced,  is  another  (in  the  shadow  pro- 
duced), since  s  is,  by  this  construction,  the  point  where  Km,  the 
line  casting  the  shadow,  pierces  the  surface  receiving  the  shadow. 
Hence  draw  srt,  and  m't  is  the  complete  boundary  of  the  shadow 
6ou<;ht.  The  shadow  of  the  hither  counterfort  is  similar,  so  far  aa 
it  i'alls  on  the  face  of  the  wall. 

Otiier  methods  of  constructing  this  shadow  may  be  devised  by 
the  student.  Let  t  be  found  by  means  of  an  auxiliary  shadow  of  um 
on  the  plane  of  the  ba8«  of  the  walL 

liemark. — In  case  an  object  has  but  few  isometrical  lines,  it  ii 
most  convenient  to  inscribe  it  in  a  right  pri>ni,  so  that  as  many  ot' 
its  edges,  as  pos>il(I(',  shrtll  lie  in  the  faces  of  the  prism. 


CHAPTER  IV. 

PBOHi.E.VS    IHVOLTTNG    THE   COXSTRUCTION   AND   EQU^i    DIVIKIOS    Ot 
CIRCLES   IX   ISOMETRICAi   DRAWING. 

220.  Prob.  9.  To  make  an  exact  constructiofi  of  the  isometrical 
drawing  of  a  circle.  PL  XIII.,  Figs.  111-112.  This  coustruction  is 
only  a  special  application  of  the  general  problem  requiring  the  con- 
struction of  ]>oints  in  the  isometric  planes. 

Let  PI.  XIII.,  Fig.  Ill,  be  a  square  by  which  a  circle  is  circum- 
scribed. The  rhombus — Fig.  112 — is  the  isometrical  di-awing  of  the; 
same  square,  CA  being  equal  to  C'A'.  The  diameters  g'h'  and  ef 
are  those  which  are  shown  in  their  real  size  at  gh  and  ef  giving 
.V,  /?,  e,  and  ^/"  as  four  pohits  of  the  isometrical  drawing  of  the  circle. 
In  Fig.  Ill,  draw  h'a\  from  the  intersection,  h\  of  the  circle  with 
A'D'  and  parallel  to  C'A'.  As  a  line  equal  to  h'a\  and  a  distance 
equal  to  K'a'  can  be  found  at  each  corner  of  Fig.  Ill,  lay  off  each 
way  from  each  corner  of  Fig.  112,  a  distance,  as  Aa,  equal  to  A'a', 
and  draw  a  line  ah  parallel  to  CA  and  note  the  point  5,  where  it 
meets  AD.  Similarly  the  points  ?i,  o,  and  r  may  be  found.  Having 
now  eight  points  of  the  ellipse  which  will  be  the  isometrical  draw- 
ing of  the  circle,  and  knowing  as  further  guides,  that  the  curve  is 
tangent  to  the  circumscribing  rhombus  at  </,  A,  e  and/,  and  perpen. 
dicular  to  its  axes  at  5,  n,  o  and  r,  this  ellipse  can  be  sketched  in  by 
hand,  or  by  an  irregular  curve. 

221.  If,  on  account  of  the  size  of  the  figure,  more  points  are  desira- 
ble they  can  readily  be  found.  Tims;  on  any  side  of  Fig.  Ill,  take 
I  distance  as  C'c'  and  c'cl  perpendicular  to  it,  and  meeting  the  circle 
.U  d' .  \\\  Fig.  112,  make  Cc^C'c',  and  make  cy/ equal  to  c'd'  and 
parallel  to  CI),  then  will  d  be  the  isometrical  position  of  the 
point  d' . 

222.  Prob.  10.  To  make  an  approximate  construction  of  the  iso 
metrical  drawing  of  a  circle.  Pl.XIII.,Fig.ll3.  By  trial  we  shall  lind 
that  an  arc,  gf  having  C  for  a  centre  and  C/for  its  radius,  will  vei-y 
nearly  pass  through  n  ;  likewise  that  an  arce/*,  with  B  for  a  centre, 
vih  very  nearly  i)ass  through  r.     These  arcs  will  be  1  angents  to  the 


100  PROIiI.EMS  IXVOLVIXG  THE  ISOMETUICAT.  DKAWING  OF  CIRCLES. 

Bides  of  the  circumscribing  square  at  tlieir  middle  points,  as  tligy 
should  be,  since  C/'and  lie  are  perpi'ndicnlar  to  these  sides  at  tlieir 
middle  [)oints.  Now  in  order  that  the  small  arcs,/'M  and  goe,  should 
be  both  tangent  to  the  former  arcs  and  to  the  lines  of  the  square  at 
<7,  h,  f  and  e,  their  centres  must  be  in  the  radii  of  the  larger  arcs, 
hence  at  their  intersections  p  and  q.  Arcs  having/)  and  q  for  cen- 
lies,  and  ph  and  qe  as  radii,  \v\\\  complete  a  four-centred  curvo 
which  will  be  a  sufficiently  near  approximation  to  the  isometrical 
ellipse,  when  tlie  figure  is  not  very  large,  or  when  the  object  for 
which  it  is  drawn  does  not  recjuire  it  to  be  very  exact. 

223.  Prob.  11.  To  make  an  isometrical  drawing  of  a  solid  com.- 
posed  of  a  short  cijlinder  cappedhrj  aliemi^phere.  ri.XlIl.,Fig.ll4. 
Scale=i.  Let  this  body  be  placed  with  its  circular  base  lowermost, 
as  shown  in  the  figure.  Make  ac  and  hd  the  height  of  the  cylin- 
drical part— 1/2  inches,  and  draw  cd-  Now  a  sphere,  however 
looked  at,  must  appear  as  a  sphere,  hence  take  e,  the  middle  point 
of  cd^  as  a  centre,  and  ec  as  a  radius,  and  describe  the  semicircle 
eAf?,  which  will  complete  the  figure. 

224.  In  respect  to  execution,  in  general,  of  the  problems  of  this 
Division,  a  descTi])tion  of  it  is  not  formally  distingtiished  from  that 
of  their  construction,  since  the  figures  generally  explain  themselves 
iu  this  respect.  In  the  present  instance,  the  visible  portion  of  the 
only  heavy  line  required  will  be  the  arc  anh.  As  there  is  no  angle 
at  the  union  of  the  hemisphere  with  the  cylinder — see  the  preceding 
probk-m — no  full  line  should  be  shown  there,  but  a  dotted  curve 
parallel  to  the  base  and  passing  through  c  and  c7,  might  be  added  to 
show  the  precise  limits  of  the  cylindrical  part. 

225.  Again,  if  it  be  desired  to  shade  this  body,  the  element,  ny^ 
of  the  cylindrical  pait,  with  the  curve  of  shade,  pfry^  on  the  si)he- 
rical  i)ar'  will  constitute  the  darkest  line  of  the  shading.  The  curve 
of  shade,  ^""'ry,  is  found  approximately  as  follows.  The  line  wy  being 
the  foremost  element  of  the  cylinder,  yy'  —  ne,  is  the  projection  of 
ftn  actual  diameter  of  the  hemisphere,  mm"  and  gg\  j)arallel  to  ST, 
are  the  radii  of  small  semiciicles  of  the  hemisphere,  to  wliich  projec. 
(ions  of  rays  of  light  may  be  drawn  tangent,  and  tn'  and  g\  are 
the  true  positions  of  their  centres — ym,'  being  equal  to  nm,  and 
1/7'  equal  to  ng.  Drawing  arcs  of  such  semicircles,  and  drawing 
rays,  fd  and  ?-S,  tangent  to  them,  we  determine  ./"and  r,  points  of 
tlie  curve  of  sh.ade  on  the  spherical  part  of  the  body,  through 
which,  with  p  and  y,  the  curve  may  be  sketched. 

226.  Viiov,.  12.    To  construct  the  isometrical  circles  on  the  thret 


PL.XIH 


pkohlkms  involving  the  isometrical  drawing  of  circles.  101 

visible  faces  of  a  mihe,  as  seen  in  an  isometrical  drawing.  PI 
XIV.,  Fig.  115.  This  figure  needs  no  minute  descnption  In  re, 
being  given  to  enable  the  student  to  become  familiar  with  the 
position  of  isometrical  circles  in  the  three  isometrical  planes,  and 
with  the  positions  of  the  centres  used  in  the  approximate  construc- 
tion of  those  circles.  By  inspection  of  the  figure,  the  following 
general  principle  may  be  deduced.  The  centres  of  the  larger  area 
are  always  in  the  obtuse  angles  of  the  rhombuses  which  represent 
the  sides  of  the  cube,  and  the  centres  of  the  smaller  arcs  are  at  the 
intersection  of  the  radii  of  the  larger  arcs  with  the  diagonals 
joining  the  acute  angles  of  the  same  rhombuses — i.  e.  the  longer 
diagonals. 

227.  Pkob.  \Z.  To  make  the  isometrical  drawing  of  a  bird  house. 
Pl.XIY.,Fig.ll6.  Assuming  C,  make  CA'  =  16  inches,  Ca=3  inches, 
and  CB  =  9  inches.  At  a,  make  aJ=:  1  inch,  and  ac= 8  inches.  Draw 
next  the  isometric  lines  BD  and  cD.  Through  b  make  i!»E=16  inches, 
make  EA  =  3  inches,  and  EF  =  7  inches.  Then  draw  the  isometric 
lines  DH  and  FH.  Bisect  5E  at  N,  make  N/=rll^  inches,  draw 
ef  and  Yf  make  ce=:FA=/>7=one  inch,  and  draw  eg  and  hg. 
Through  A  and  b  draw  isometric  lines  which  will  meet,  as  at  a'. 
On  JE  make  bJc,  hn  and  wE  each  equal  to  3  inches,  and  let  kl  and 
mn  each  be  3j  inches.  At  I  and  ?^  draw  lines,  as  ?y,  parallel  to  CB, 
and  one  inch  long,  and  at  their  inner  extremities  erect  perpendicu- 
lars, each  3^  inches  long.  Also  at  A,  ?,  m  and  n^  draw  vertical  iso- 
metrical Hues,  as  kt^  3|-  inches  long.  The  rectangular  openings  thus 
formed  are  to  be  completed  with  semicircles  whose  real  radius  is 
If  inches,  hence  produce  the  lines,  as  kt — on  both  Avindows — mak- 
ing lines,  as  /cG,  5\  inches  long,  and  join  their  upper  extremities  a3 
at  GI.  The  horizontal  lines,  as  ts,  give  a  centre,  as  5,  for  a  larger 
aic,  as  tu.  The  intersection  of  Go  with  Jz — see  the  same  letters  on 
E'ig.  115 — gives  the  centre,  jt?,  of  the  small  arc,  uo.  The  same  ope- 
rations on  both  openings  make  their  front  edges  complete.  Make 
oq  and  j>r  parallel,  and  each,  one  inch  long,  and  r  will  be  the  centre 
of  a  small  arc  from  q  Avhich  forms  the  visible  part  of  the  inner  edge 
of  the  window.  Suppose  the  corners  of  the  platform  to  be  rounded 
by  quadrants  whose  real  radius  is  14-  inches.  The  lines  a'b  and 
bk  each  being  3  inches,  k  is  the  centre  for  the  arc  which  repre- 
sents the  isometric  drawing  of  this  quadrant,  whose  real  centre 
on  the  object,  is  indicated  on  the  drawing  at  y.  So,  near  A,  w  ia 
the  centre  used  in  drawing  an  arc,  which  represents  a  quadrant 
vhose  centre  is  x. — See  the  same  letters  on  Fig  115. 


102  PROBLEMS  IXVOLVIXG  THE  ISOMETRICAL  PRAWING  OF  CIRCLES. 

Of  the  Isometrical  Draicing  of  Circles  which  are  divided  in 
Equal  Parts. 

228.  Prob.  14.  PI.  XIV.,  Fig.  117.  First  method.— l^  tlie  semi 
ellipse,  ADB,  be  revolved  up  into  a  vertical  position  about  AB  as  an 
axis,  it  will  appear  as  a  semicircle  AD'B  of  which  ADB  is  the  iso- 
metrical projection.  Since  AB,  the  axis,  is  parallel  to  the  vertical 
plane,  the  arc  in  which  any  i)oint,  as  D,  revolves,  is  in  a  plane  per 
pendicular  to  the  vertical  plane,  and  is  therefore  projected  in  a 
straight  line  DD'.  Hence  to  divide  the  semi-ellipse  ADB  into  parts 
corresponding  to  the  parts  of  the  circle  which  it  represents,  divide 
AD'B  into  the  required  number  of  equal  parts,  and  through  the 
points  thus  found,  draw  lines  parallel  to  D'D,  and  they  will  divide 
ADB  in  the  manner  required.  The  opposite  half  of  the  curve  can 
of  course  be  divided  in  a  similar  manner. 

229.  Second  method. — CE  is  the  true  diameter  of  the  circle  ot 
which  ADB  is  the  isometrical  drawing.  Let  it  also  represent  the 
side  of  the  square  in  which  the  original  circle  to  be  drawn  is  inscribed. 
The  centre  of  this  circle  is  in  the  centre  of  the  square,  hence  at  O, 
found  by  making  eO  equal  to  half  of  CE,  and  perpendicular  to  that 
line  at  its  middle  point  e. 

With  O  as  a  centre,  draw  a  quarter  circle,  limited  by  CO  and  EO, 
and  divide  it  into  the  required  number  of  parts.  Through  the  points 
of  division,  draw  radii  and  produce  them  till  they  meet  CE.  CE, 
considered  as  the  side  of  the  isometrical  drawing  of  the  square,  is 
the  drawing  of  the  original  side  CE  of  the  square  itself  with  all 
its  points  1,  2, C,  7,  &c.,  and  O'  is  the  isometrical  posi- 
tion of  O.  Hence  connect  the  points  on  CE  with  the  point  0'  and 
the  lines  thus  made  will  divide  the  quadrant  BC  in  the  manner 
required. 

Aj^lications  of  the  preceding  Problem,. 

230.  Prob.  15.  To  make  an  isometric  drawing  of  a  segment  oj 
an  Ionic  Column.  PI.  XIV.,  Fig.  118.  Let  aD  be  a  side  of  the  cir 
cumscribing  piism  of  the  column.  By  the  second  method  of  Prob. 
14,  fmd  O',  the  centre  of  a  section  of  the  column,  and  with  O'  as  a 
centre,  draw  any  arc,  as  a'q'.  The  curved  recesses  in  the  surface 
of  a  column  are  called  flutes,  or  the  colunm  is  said  to  be  fluted.  In 
an  Ionic,  and  in  some  other  styles  of  cohmms,  the  flutings  are  semi- 
circular with  narrow  flat,  or  strictly,  cylindrical  surfaces,  as  ef"/), 
between  them.  Hence,  in  Fig.  118,  assume  a'b\  equal  to  q'v\  aa 
half  of  a  space  between  two  flutes,  divide  h'v'  into  four  equal  parts, 
and  make  the  points  of  division  central  points  of  the  spaces  as  f'e' 


PROBLEMS  INVOLVING  THE  ISOMETRICAL  DRAWING  OP  CIRCLES.  103 

between  the  flutes.  Lei  the  flutes  be  drawn  witli  points,  as  c'  a» 
centres  and  touching  the  points  as  b'cl ;  then  draw  an  arc  tangent, 
as  at  r,  to  the  flutes.  To  proceed  now  with  the  isometrical  draw- 
ing, draw,  in  the  usual  way,  the  isoraetiical  drawing  of  the  outer 
circumferences  of  the  column,  tangent  to  aD  and  h"'¥ — assuming 
DF  for  tlie  tliickness  of  the  segment.  Now  a'q'  being  any  arc,  and 
not  one  tangent  to  oD  so  as  to  represent  the  true  size  of  a  quadrant 
of  the  outer  circumference,  the  true  radius  of  the  circle  tangent  to 
the  inner  points  of  all  the  flutes  will  be  a  fourth  proportional,  O'y' 
to  0/',  Oi  (=^0'y),  and  O's.  On  O^,  lay  off  OY=Oy',  draw  IJ  to 
find  a  centre  I,  and  similarly  find  the  other  centres  of  the  larger  area 
of  the  inner  ellipse.  The  points  ?i,  h  and  n\  h'  are  the  centres  of  the 
small  arcs  (222)  for  the  two  bases.  Having  gone  thus  far,  produce 
0'b\  O'c',  &c.  to  dD  ;  at  i,  c,  &c.,  erect  vertical  lines,  hh"\  cc"\ 
&c.,  then  from  6,  c,  &c.  draw  lines  to  O,  and  note  their  intersec- 
tions, h'\  c",  &c.  with  the  curves  of  the  low'er  base ;  and  from 
h"\  c"\  Sec.  draw  lines  to  O"  and  note  tlieir  intersections,  h"'\  c"'\ 
&c.  with  the  elHpses  of  the  upper  base.  This  process  gives  three 
points  for  each  flute  by  which  they  can  be  accurately  sketched  in, 
remembering  that  they  are  tangent  to  tlie  inner  dotted  ellipses,  as 
at  c"",  o"\  &c.  and  to  the  radii,  as  c"0" — at  e" .  Parts  beyond 
FO"  are  projected  over  from  the  parts  this  side,  thus  drawn. 

231.  Prob,  16.  To  construct  the  isometrical  drawing  of  a  seg- 
ment of  a  Doric  Coliunn.  PL  XIV.,  Fig.  119.  The  flutes  of  a  Doric 
column  are  shallow  and  have  no  flat  space  between  them.  Adopt- 
ing the  first  method  of  Prob.  14,  let  the  centre,  A,  of  the  plan  be 
in  the  vertical  axis,  GA',  of  the  elevation,  produced.  Let  Ac  and 
Ab  be  the  outer  and  inner  radii  containing  points  of  the  flutes. 
Make  Ad=^  of  Ac,  for  the  radius  of  the  circle  which  shall  contain 
the  centres  of  the  flute  arcs.  Let  thei'e  be  four  flutes  in  the  qua- 
drant, shown  in  the  plan.  Their  centres  will  be  .at  h,  &c.,  where 
radii  A^,  &c.,  bisecting  the  flutes,  meet  the  outermost  arc.  In  pro 
ceeding  to  construct  the  isometrical  drawing,  project  b  and  c,  at  b 
and  c'  on  the  axis  A'd'.  Now,  owing  to  the  variation  at  b  and  c' 
between  the  true  and  the  approximate  ellipse,  we  cannot  make  use 
of  the  latter,  if  we  retain  b'  and  c  in  their  proper  places,  as  projected 
from  b  and  c,  hence  through  b'  and  c'  draw  isometric  lines  which 
locate  the  points  N'  and  Q'  (the  points  are  between  these  letters) 
which  are  the  true  positions  of  N  and  Q  respectively.  Correspond- 
ing points,  between  N'"  and  v,  are  similarly  found.  By  an  irregular 
cu>"ve  the  semi-ellipses  vb'Q,'  and  N"V''N'   can  he  quite  accurately 


104peoblems  in^"Oltixg  the  isometeical  dkawixg  of  circles- 

drawn.  Next,  project  upon  these  curves  the  points  u^  e,  &c.,  r,  g^ 
tfcc.  of  the  flutes — as  at  u\  e\  &c.,  r',  <fec.,  and  Avith  an  irregular 
curve  draw  the  curves  through  these  points,  tangent  to  the  inner 
semi-ellipse.  The  corresponding  curves  of  the  lower  base  aie  found 
by  drawing  lines  r'r'\  ic'u",  tfcc.  through  the  points  of  tangency, 
r',  k\  &c.,  and  through  w',  &c.,  and  all  equal  to  FD,  tlie  thickness 
of  the  segment.  The,  curves  above  the  axis  A'd'  are  projected 
across  from  those  already  made  below  it.  Let  this  figure  and  the 
last  be  made  separately  on  a  very  large  scale, 

Special  Examples. 

232.  Prob.  17.  To  draw  a  cube  or  other  paraJMopipedical  body 
to  as  to  show  its  under  side.  PI.  XIV.,  Fig.  120.  By  reflection,  it 
becomes  evident  that  it  is  the  relative  direction  of  the  lines  of 
the  drawing  among  themselves,  that  make  it  an  isometrical  draw- 
ing. Hence  in  the  figure,  where  all  the  lines  are  isometric  lines,  the 
whole  is  an  isometric  drawing,  now  that  the  solid  angle  C  is  nearest 
us,  as  much  as  if  the  angle  A  (lettered  C  on  previous  figures)  were 
nearest  us. 

233.  Remark.  By  a  curious  exercise  of  the  will,  we  can  make 
Fig.  120  appear  as  an  interior  view,  showing  a  floor  CFED,  and 
two  Avails;  or,  in  Fig.  115  and  others,  we  can  picture  to  ourselvag 
an  interior  showing  a  ceiling  GMz  and  two  Avails.  This  is  probably 
because — \st.  All  drawings  being  of  themselves  only  plane  figures, 
we  educate  the  eye  to  see  in  them,  what  the  mind  chooses  to  conceive 
of,  as  having  three  dimensions.  Ind.  When,  as  in  isometrical  draw- 
ing, the  draAving  in  itself  as  a  plane  figure,  is  the  same  for  an  ulterior 
as  for  an  exterior  view  of  any  given  magnitude,  the  eye  sees  in  it 
whichever  of  these  two  the  mind  chooses  to  imagine. 

234.  Prob.  18.  To  construct  isometrical  drawir^ys  of  oblique 
sections  of  a  right  cylinder  with  a  circxdar  base.  PL  XIV.,  I'ig. 
121.  This  construction  is  easily  made  from  a  given  circle  as  a  base 
of  the  cylinder,  that  base  being  in  an  isometric  plane.  The  circle  in 
the  plane  AGEF  is  such  a  circle.  Let  A'G'E'F'  be  a  plane  inclined 
to  AGEF  but  perpendicular,  as  the  latter  is,  to  the  planes  GB  and 
DF,  and  let  A"G''E''F''  be  a  plane  inclined  to  all  the  sides  of  the 
prism  AGE — D. 

Lines,  as  aa'a\  <fec.,  being  in  the  faces  of  the  prism  and  parallel 
to  their  edges,  meet  the  intersections,  F'E' — F^E",  &c.  of  the 
oblique  planes  at  points  a',  a',  etc.,  which  are  points  of  oblique 
sections  of  a  cylinder  inscribed  in  the  prism  AGE — D,  and  wQose 
base  is  acbdu. 


PROBLEAIS  IXYOLVING  THE  ISOMETKICAL  DRAAVIXG  OF  CIRCI-KS.  105 

So,  points,  as  c,  >have  the  corresponding  points  c'c\  <fec.  in  the 
diagonals  A'E',  A'E"  of  tlie  planes  in  which  those  points  are  found. 

To  find  points,  as  t\  t'\  etc.  corresponding  to  t  in  the  base,  draw 
any  hne,  as  yo?,  through  t,  and  find  the  corresponding  lines,  as  y'cV 
and  y''d".  Their  intersections  with  the  diagonals  G'F'  and  G"F' 
:vill  give  the  points  t\  t\  &c.  Plaving  thus  found  eight  points  of 
f-acb  ohlique  section  of  the  given  inscribed  cylinder  whose  base  is 
ahcd-ii^  and  lemembering  that  each  of  these  sections  is  tangent  to  the 
sides  of  its  circumscribing  polygon  (considering  the  lines  y'd',&G.)j 
the  curves  a',  b,'  c',  t',  and  a",  b",  c",  t"  are  readily  sketched  in. 

235.  Remarks,  a.  As  before  stated,  it  is  the  relative  direction, 
among  themselves,  of  the  lines  of  an  isometrical  drawing,  that  deter- 
mine it  as  an  isometrical  drawing,  hence  PI.  XIV.,  Fig.  121,  is  an 
isometrical  drawing,  though  its  lines  are  not  situated  with  refer- 
ence to  the  edges  of  the  plate  as  the  similar  lines  of  previous 
figures  have  been.  If  the  portion  of  the  plate  containing  this  figure 
were  cut  out  so  as  to  make  the  edges  of  the  fragment,  so  cut  out, 
parallel  and  perpendicular  to  GE,  the  figure  would  appear  like 
the  previous  isometrical  drawings. 

h.  The  problem  just  solved  must  not  be  confounded  with  one 
which  should  seek  to  find  the  isometric  projection  of  a  curve 
which  in  space  is  a  circle  on  the  plane  G'E' — A',  for  the  curve 
a'b'c'd't'  is  not  a  circle,  in  space. 

236.  Peob.  19.  To  solve  the  problem  just  enunciated.  PI.  XIV., 
Figs.  121-122.  e"r — Fig.  122— is  a  plan  of  the  section  rA'F'  in 
which — it  being  a  square — a  circle  can  be  inscribed.  e"r  is  there- 
fore the  plan  of  the  circle  also.  Making  rG — Fig.  122 — equal  to 
rG' — -Fig.  121,  and  drawing  e"G,  we  have  the  plan  of  the  section 
G'E' — A',  and  making  o"p>\  Fig.  122,  equal  to  eV,  we  have  the  plan 
of  a  circle  in  the  section  G'E' — A'.  Now  draw  o''x  and  ju'e' — Fig. 
122 — make  A'e  and  G'e'"  and  e"'p  and  eo — Fig.  121 — equal  to  e'a;, 
Ge',  e'p'  and  xo" — Fig.  122 — draw  ji:>Y  and  oU;  and  w'lJ  and  b'Y 
to  intersect  them,  and  we  shall  have  U  and  Y  as  the  isometric 
positions  in  the  plane  G'E' — A'  of  the  points  o'  and  p'  which, 
considered  as  points  on  the  circle,  are  evidently  enough  extre- 
mities of  its  horizontal  diameter,  at  which  points,  the  circle  is  tan- 
gent to  the  vertical  lines  whose  isometric  positions  in  the  piano 
G'E' — A'  are7:)Y  and  oU.     T  and  a'  are  other  points. 

The  finding  of  intermediate  points,  which  is  not  difficult,  is  left  ad 
an  exercise  for  the  student. 


CHAPTER  V. 


OBLIQUE    PROJECTIONS. 


A.  There  is  a  kind  of  projection,  examples  of  which,  in  the  draw 
ing  of  details,  etc.,  are  oftener  seen  in  French  works  than  isometri 
cal  projection  (an  English  invention)  is.  It  has  been  variously 
named,  "  Military,"  "  Cavalier,"  or  "  Mechanical "  Perspective.  It 
may  be  called  "  Cabinet  Projection,"  it  being  especially  applicable 
to  objects  no  larger  than  those  of  cabinet  work,  and  being  actually 
used  in  representing  such  work.  It  is  properly  called  oblique 
projection,  because  in  it  the  projecting  lines,  which  have  been 
hitherto  made  perpendicular  to  the  plane  of  projection  are  oblique 
to  that  plane;  and  ^lictorial  projection^  on  account  of  its  pictorial 
effect,  as  seen  in  PI.  I.,  Fiirs.  1,  2,  3,  etc. 

2.  This  new  projection  differs  from  isometrical,  chiefly  in  show- 
ing two  of  the  tliree  dimensions  of  a  cube,  for  example,  in  their  true 
direction  as  well  as  size. 

Thus;  Fig.  1  is  the  isometrical  drawing  of  a  cube,  and  Fig.  2  is 


/ 

/\ 

/\ 

h[ 

c 

'"/{ 

/ 

0 

Fig.  2. 


an  oljiifjue  })rojection  of  the  same  cube  ;  all  the  edges  being  of  the 
same  length  in  both  figures.  Hence  we  see,  as  stated,  that  in  the 
latter  figure,  the  faces  DEFG,  and  ABCII,  and  by  consequence 
every  line  in  them,  are  shown  in  tlieir  true  form,  as  well  as  sIm  ^ 
which  is  not  true  of  isometrical  drawing. 


OBLIQUE   PROJECTION'S.  107 

3.  Another  advantage  of  oblique  projection,  already  apparent,  is, 
that  the  remote  corner,  H,  which,  in  the  isonietrical  drawing  of  a 
cube,  coincides  with  the  foremost  corner,  G,  is  seen  separately  in 
the  oblique  piojection. 

Also,  of  the  four  body  diagonals  of  the  cube,  one,  GIT,  appears 
as  a  point  only,  in  i.sometrical  projection,  and  the  other  three,  as 
FC,  are  all  partly  confounded  with  the  projections,  as  FG,  of  edges. 
But  in  the  oblique  projection,  all  these  diagonals  show  as  lines,  and, 
except  BE,  separately  from  the  edges  of  the  cube. 

4.  In  the  projections  hitherto  considered,  the  projecting  lines  of 
a  point  have  uniformly  been  taken  perpendicular  to  the  planes  of 
projection. 

Sometimes,  however,  the  projecting  lines,  or  direction  of  vision 
(12)  are  oblique  to  the  plane  of  projection. 

There  are  thus  two  systems  of  projection  in  which  the  eye  is  at 
an  infinite  distance,  l^lrst,  common  or  peiyendicular  p7'oJections, 
in  which  the  projecting  lines  (5)  are  perpendicular  to  the  plane  of 
projection.  Second.  Oblique  projections^  in  which  those  lines  are 
oblique  to  the  plane  of  projection, 

Isometrical,  is  a  species  of  perpendicular  projection.  We  shall 
now  proceed  to  ex2Dlain  the  simple  and  useful  form  of  projection 
which  is  called  oblique. projection. 

5.  If  a  line,  AB,  Figs.  3,  or  3a,  be  perpendicular  to  any  plane 
PQ,  its  projection  on  that  plane,  in  common  projection,  would  be 
simply  the  point  B.  But  if  we  suppose  the  projecting  line,  AC,  of 
any  point.  A,  to  make  an  angle  of  45°  with  the  plane  of  projection 
PQ,  it  is  evident  that  the  piojection  of  A  would  be  at  C,  and  the 
projection  of  AB  on  PQ  would  be  BC;  also  that  BC=AB.  That 
is,  the  projectioti,  as  BC,  of  a  j^eipendicular  to  the  plane  of  project 
tion  is  equal  to  that  perpendicular  itself. 

Any  line  through  A  and /)a?'a/^eZ' to  the  plane  PQ  would  evidently 
be  projected  in  its  real  size  on  PQ.  Hence,  finally,  the  system  of 
oblique  ])rojections  here  described,  allows  us  to  show  the  three 
dimensions  of  a  solid  in  their  real  size^  on  a  single  figure;  but  only 
parallels.,  and  perpendicidars  to  the  plane  of  projection,  appear  in 
their  true  size. 

6.  It  is  now  evident  from  Fig.  3a,  that  there  may  be  an  infinite 
number  of  lines  from  A,  each  making  an  angle  of  45°  with  the 
plane  PQ.  Tnese  lines,  taken  together,  would  form  a  right  cone 
with  a  circular  base,  whose  axis  would  be  AB,  whose  vertex  would 
be  A,  and  whose  base  would  be  a  circle  in  the  plane  PQ,  drawn 
with  B  as  a  centre,  and  BC  as  a  radius.     Each  radius  of  this  circle 


108 


OBLIQUE   PROJECTIONS. 


would  be  an  oblique  projection  of  AB,  corresponding  willi  the 
element  as  C'A,  from  its  extremity,  taken  as  the  direction  of  the 
projecting  lines.  That  id,  the  ohliqiie  projection  of  AB  may  be 
drrfrin  equal  to  AB  and  in  any  direction. 

7  Thus  Figs. 4,  h,  5  and  many  more  are  all  equally  oblique  projec- 
tions of  the  same  c  ibe.  The  paper  represents  the  plane  of  projec- 
tion ;  FA  is  perpt.iditmlar  to  the  paper  at  A.  The  eye,  relative  tc 
Fig.  2,  is  looking /roy.''  'n,  infinite  distance  above  and  to  the  right 
of  the  body,  and  in  a  direction  making  an  angle  of  45°  with  the 
pa}>er.  And,  generally,  in  oblique  projection,  the  direction  of 
Xiinon^^the projecting  lines^  may  have  any  direction  (the  same  for 
ail  points  in  the  same  problem)  making  an  angle  of  <^b'^  with  the 
jilane  of  projection.  Ilence  FA  may  have  any  direction  relative 
tc»  FG  and  y(it  be  always  equal  to  FG,  that  is  to  the  original  of  FA 
.n  space.  Thus,  CDK  =  45°,  in  Fig.  4 ;  30°,  in  Fig.  5;  and  60°,  in 
Fig.  6.     Also,  DC,  FA,  etc.,  may  incline  to  the  left^  or  downward. 


Fig.  6. 


OBLIQUE    PROJECTION'S. 


109 


In  Fig.  G,  CDK  =  60°.  Accordingly  DK==iDC,  from  wliich, 
having  ibund  K,  the  perpendicular  KC  can  be  drawn  to  limit  DC. 
Or,  as  bef«»re,  CK=i  Vs^  DC  being  =1.  That  is,  CK=r-iEC,  since 
CDE=:12(>°.  And  for  a  square  prism  of  any  length,  KC=  half  of 
the  diagonal  joining  alternate  vertices  of  a  regular  hexagon  whose 
side  equals  the  edge  of  the  prism,  lying  in  the  direction  of  DC,  an  J 
whose  length  is  supposed  to  be  given. 


Fig.  6. 

With  these  illustrations,  the  student  might  proceed  to  investi- 
gate other  relations  between  the  parts  of  these,  and  still  other 
oblique  projections.     But  the  above  may  suffice  for  now. 

We  observe  that,  in  Figs  5  and  6,  none  of  the  body  diagonals 
are  confounded  with  the  edges ;  and  that  each  of  the  tJiree  forma 
may  be  preferable  for  certain  objects. 

8.  Points  not  on  the  axes  EF,  EH,  and  ED,  or  on  parallels  to 
them,  are  found  by  co-ordinates,  as  in  isometrical  drawing.  Thus, 
if  ED,  Fig.  4,  be  4  inches,  and  if  we  make  Ea  =  2  inches,  ab  paral- 
lel to  EH,  =  2|-  inches,  and  be,  parallel  to  EF,=::1|-  inches,  then  c  is 
the  oblique  projection  of  a  point,  2  inches  fi-oui  the  face  FH  ;  2^ 
inches  from  the  face  FEG,  and  l^  inches  above  the  base  EDC. 
This  principle  will  enable  the  student  to  reconstruct  any  of  the 
preceding  isometrical  examples  of  straight-edged  objects,  in  ob- 
lique projection. 

9.  It  only  now  remains  to  explain  the  oblique  projections  of  cir- 
cles. Let  Fig.  7  be  the  oblique  projection  of  a  cube,  with  circles 
Inscribed  in  its  three  visible  faces.  One  of  these  circles,  abed,  will 
appear  as  a  circle,  and  so  would  the  invisible  one  on  the  parallel 
rear  face. 

For  the  ellipse  in  BCDG,  draw  the  diagonals,  BD  and  CG,  ol 


no 


OBLIQUE    PROJECTIONS. 


that  face.  Then  in  the  cube  itself,  horizontal  lines  joining  corrO' 
epondin;^  points  in  the  circles  abcd^  and  hpfu^  are  parallel  to  the 
diagonal  EC.    Hence  tu^  ef,  mp^  and  gh  determine  the  points  u^ftP 


Pig.  7. 

and  A,  by  their  intersections  with  the  diagonals  BD  and  CG.  Tlie 
middle  points  of  the  sides  of  the  fice  BCDG,  are  also  points  of  the 
ellipse,  and  are  its  points  of  contact  with  those  sides.  The  ellipse 
also  has  tangents  at  h  and/",  paiallcl  to  BD,  and  at  u  and  jy^  paral- 
lel to  CG.  Hence,  having  eight  points,  all  of  which  are  ])oints  of 
contact  of  known  tangents,  the  ellipse  can  be  accurately  sketched. 

10.  The  ellipse  in  the  upper  face  could  be  found  in  the  same 
manner.  But  an  approximate  construction  by  circular  arcs  has 
been  sliown,  to  test  its  accuiacy  and  appearance,  as  compared  with 
the  approximate  isometrical  ellipse.  The  ellipse  being  tangent  at 
L  and  K,  perpendiculars  to  FA  and  BA,  at  those  points,  will  in- 
tersect at  M,  the  centre  of  an  arc  tangent  to  FA  and  BA  at  those 
points.  Then  «N  pei|)en(licidar  to  FG  at  a,  and  equal  to  ]\IK, 
gives  N,  the  centre  of  the  arc  an.  As  the  remaining  arcs  must  bo 
tangent  to  those  just  drawn,  their  centres,  r  and  «,  must  be  the 
intersections  of  the  radii  of  the  large  arcs,  with  the  transverse  axis, 
KM,  of"  the  ellijise. 

The  true  extremities  of  the  transverse  axis  are  found  by  drawiny 


OBLIQUE    PROJECTIONS.  Ill 

pq  parallel  to  AC,  and  a  parallel  to  it  from  %c.  Tlie  error  qq'  al 
each  end  of  the  transverse  axis,  is  thus  seen  to  be  considerable 
Also  the  jTieater  difference  between  the  radii,  than  occurs  in  niak- 
ins:  the  isonietrical  ellipse,  occasions  a  harsh  change  of  curvature  at 
K^  N,  etc. ;  so  that  the  approximate  construction  of  the  oblique 
ellipse  is  of  very  little  value. 

11.  It  is  found  on  trial,  that  the  centre  M  falls  both  on  BD  and 
EC,  so  that  neither  KM  nor  LM  really  need  be  drawn.  The  reason 
of  this  property,  Avhich  so  simplifies  the  construction,  is  evident. 
For  BAmAF^^FE  are  in  position  as  three  sides  of  a  regular  octa- 
gon, so  that  the  p  .rpendiculars,  as  KM,  from  the  middle  points  of 
those  sides,  will  ueet  at  the  same  points  with  BD,  AGM,  EC,  etc., 
which  are  obviously  the  bisecting  lines  of  the  angles  of  the  octa 
gon,  viz.  at  the  centre,  M,  of  the  octagon. 

By  varying  the  angle  GFA,  as  in  the  previous  figures,  the  stu- 
dent may  discover  similar  coincidences,  which  he  can  explain  for 
himself. 

Finally,  it  is  to  be  noticed,  that  the  pictorial  diagrams  of  PI.  I., 
Figs.  1,  2,  3,  5,  etc.,  which  are  so  effective  a  substitute  for  actual 
models,  to  most  eyes,  are  merely  oblique  projections  of  models 
themselves. 

Practical  Examples. 

12.  PI.  XV.  shows  some  further  illustrations  of  oblique  projec- 
tion in  contrast  with  isometrical  drawing. 

Fig.  1  is  an  isometi'ical  drawing  of  a  roller  and  axle,  showing  the 
parallel  circumscribing  squares  of  its  several  parallel  circles;  and 
the  circumscribing  prism,  mnpo,  of  the  roller,  placed  so  as  to  show 
its  lower  base.  The  distances  ah^  bd  and  de,  between  the  ccntrea 
of  the  circles,  are  thus  seen  in  their  true  size,  in  this,  and  on  the 
next  two  figures.  Also  the  several  circles  and  their  centres,  have 
the  same  letters  on  the  same  figures. 

Fig.  2  shows  an  oblique  projection  of  the  same  object,  when  its 
axis,  «e,  is  made  perpendicular  to  the  paper.  This  is  the  simplest 
position  to  give  to  the  object,  since  its  several  circles,  being  then 
parallel  to  the  paper,  will  appear  respectively  as  equal  circles  in 
the  figure.  And,  generally,  in  making  oblique  projections  of  ob- 
jects having  some  circular  outlines,  the  object  should  be  so  placed, 
that  the  majority  of  these  outlines  should  be  in  planes  parallel  to 
Ihe  plane  of  projection. 

Example. — Make  ae  in  any  other  direction. 

Fig.  3  shows  another  oblique  projection  of  the  samo  object,  but 


112  OBLIQUE    PROJECTIOXS. 

witli  its  axis  ae  parallel  to  the  paper,  or  plane  of  projection.  Dif- 
fort-nt  wheels  and  their  axles  in  the  same  machine,  might  have  the 
two  positions  indicated  in  Figs.  2  and  3.  Hence  it  is  necessary  to 
understand  both;  thoiigh  if  drawing  only  a  single  object  of  this 
kind,  we  should  for  convenience  make  it  as  in  Fig.  2,  only  remem- 
bering, as  explained  in  previous  principles,  that  ae  may  be  drawn 
in  any  direction. 

Examples. — Is^.  In  Fig.  1,  let  the  upper  end  of  the  axis  be  the 
visible  one. 

2d.  In  Fig.  3,  let  ae  be  horizo)ital  and  parallel  to  the  paper  and 
let  the  left  hand  end  of  the  body  be  seen. 

PI.  XV.,  Figs.  4,  5,  and  G  are  a  plan  and  two  isonietrical  draw- 
ings, in  full  size,  of  a  hexagonal  nut.  Fig.  4  is  the  plan  of  the  nut 
with  the  circumscribing  rectangle,  ninop,  containing  two  of  its  sides, 
CD  and  AF.  Fig.  5  is  the  isonietrical  drawing  of  the  same,  and 
thus  shows  the  face  CD/i,  in  its  true  size.  The  edges  B^i,  Cc,  etc., 
and  centre  heights,  IIA,  of  the  faces,  also  show  in  their  real  size,  as 
does  the  height  Oo  of  the  nut.  BC  is  greater  than  its  real  size, 
being  more  nearly  parallel  to  pn  than  pm  is.  AB  is  less  than  its 
true  size,  AB,  Fig.  4;  being  nearer  perpendicular  to  pn  than  pm  is. 

Fig.  6  is,  peihaps,  a  more  agreeable  looking  isometrieal  figure 
of  the  nut,  but  it  shows  only  the  heights  in  their  true  sizes,  except 
an  jyq  equals  the  diameter  of  the  circumscribing  circle  MNL  of  the 
hexagonal  base  of  the  nut,  so  that  half  of  ^:>2'  equals  the  true  width 
of  ilie  faces. 

This  figure  makes  an  application  of  (Frob.  XIV.,  J^irst  Method). 
Thus,  having  made  the  isometrieal  circle  MNKL,  in  the  usual  way, 
describe  the  semicircle  MN'K'L  on  ML  as  a  diameter,  and  inscribe 
the  semi-hexagon  in  it  with  vertices  as  M,  N',  K'  and  L.  Then 
by  revolving  the  semicircle  back  to  its  isometrieal  position,  N' and 
K'  will  fall  at  N  and  K. 

The  surface  Q,  in  Figs.  5  and  G,  represents  a  spherically  rounded 
surface  of  the  nut,  while  the  surface,  R,  is  plane.  By  finding  three 
points  as  c,  A  and  D,  Fig.  5,  in  each  upper  edge  of  a  face,  those 
edges  can  be  drawn  as  circular  arcs;  and  the  visible  boundaries, 
grj,  of  the  rounded  surface  Q  can  be  sketched,  as  indicated. 

Examples. — 1*^^.  Make  oblique  p)'>'oJections  corresponding  to  Figs. 
5  and  G. 

2d.  Also  with  the  top>,  R,  of  the  uwt  2yarallel  to  the  plane  of  pro- 
jection,  and  either  in  isometrieal  or  oblique  projection. 

Sd.  Also  as  if  Fig.  0  were  turned  90°  about  Oo,  8(t  as  to  show 
only  two  faces  of  the  nut. 


OBLIQUE    PROJECTIONS.  113 

Figs.  7,  8,  and  9  show  a  plan  and  oblique  projection  of  a  ra(;de) 
of  an  oblique  joint. 

Fig.  V  shows,  once  for  all,  that  in  every  case  of  ohlique.,  as  wel) 
as  of  isometrical  drawing,  where  the  lines  as  dg  and  gp^  of  the  ob- 
ject, are  oblique  to  each  other,  the  body  must  be  conceived  to  be 
inclosed  in  a  circumscribing  rectangular  prism,  whose  sides  shall 
contain  its  points,  or  from  which  they  can  be  laid  off  by  ordinatea 
as  mo,  parallel  or  perpendicular  to  those  sides. 

Fig.  7  is  on  a  scale  of  one  half,  and  Fig.  9  is  in  full  size.  Then, 
supposing  the  scale  to  be  the  same  Cw,  Fig.  9,  =  en  Fig.  7,  mo. 
?iP,  ihy  etc.,  in  Fig.  9  =  mo,  np^  ih,  etc.,  in  Fig.  7.  So  fe  and /a 
Fig.  9  =  the  same  in  Fig.  7. 

Thus  the  edges  of  timber  A  are  shown  in  their  real  size,  but 
those  of  B  are  distorted  by  their  position.  B  is  separately  shown 
in  its  true  proportions  in  Fig.  8,  that  is  so  far  as  its  Hoes  arc  paral- 
lel to  ^0,  oO  or  op. 

Of  the  heavy  lines  in  Ohlique  Projection. 

These  simply  follow  the  same  rule,  relative  to  the  given  object 
that  is  applied  in  common,  or  perpendicular  projections;  (lG-20). 

Thus,  in  PL  XV.,  Fig.  2,  the  semicircles  of  A,  B  and  C  below 
ae  would  be  heavy,  and  the  opposite  parts  of  D,  and  E.  Also  if  B, 
Fig.  8,  represents  a  timber  parallel  to  the  ground  line,  the  heavy 
lines  would  be  as  there  shown.  And  likewise  on  Fig.  9,  where 
these  lines  are  indicated  by  double  dashes  across  them. 

In  short,  conceive  the  common  projections  of  an  object  to  be 
given  with  the  heavy  lines  drawn.  The  oblique  projection  of  the 
same  object,  placed  in  the  same  position,  would  simply  show  the 
oblique  projections  of  the  same  heavy  lines.  That  is,  the  same 
lines  would  be  heavy  in  both  kinds  of  projection. 


DIVISION     FIFTH. 

ELEMENTS  OP  MACHINES. 


CHAPTER  I. 

PRINCIPLES.      SUPPORTEllS  AND   CRANK  MOTIONS. 

General  Ideas. 

1.  Machines  generally  effect  only  jo/iyszcaZ  changes.  That  is, 
they  are  designed  to  change  either  the  form  or  the  position  of 
matter.  They  do  this  either  directly,  as  in  machines  that  ope- 
rate immediately  on  the  raw  material  to  be  wrought,  as  looms, 
lathes,  planers  of  wood  or  metal,  etc.,  or  indirectly,  as  in  the 
machines  called  prime  movers,  like  steam-engines  and  water- 
wheels,  which  actuate  operating  machines. 

We  have,  then.  Prime  movers  and  Operative  machines.  Also, 
of  the  latter,  machines  for  changing  the  position  of  matter,  as 
pumps,  cranes,  etc.;  and  machines  for  changing  its  form,  as 
lathes,  planers,  etc. ;  and  each  with  many  subdivisions. 

2.  In  every  machine  there  are  to  be  distinguished  the  sup- 
porting parts,  which  are  generally  fixed  and  rigid,  and  the 
working  parts,  which  are  moving  pieces. 

The  supporting  parts  are  general,  supporting  the  entire  ma- 
cliine;  or  local,  supporting  some  one  part,  as  the  pillow -Mock,  also 
called  a  plummer-block,  or  a  pedestal,  which  supports  a  revolv- 
ing shaft ;  or  the  guide  bars,  plainly  seen  in  some  form  at  the 
piston-rod  end  of  the  cylinder  of  any  locomotive  or  other  steam- 
engine,  and  which,  by  means  of  the  stout  block,  called  a  cross- 
head,  sliding  between  them,  constrain  the  piston-rod,  which  is 
fastened  to  the  cross-head,  to  move  in  a  straight  line. 

3.  77ie  working  parts  are  connected  together,  forming  a  train, 
subject  to  this  law,  that  a  given  position  of  any  one  piece  deter- 
mines that  of  all  the  others.  For  the  purpose  of  making  the 
drawing  of  a  machine,  it  is  not  enough,  tliercfore,  only  to  take 


PL.  XIV. 


c 


SUPPOBTEES   AND   CRANK   MOTIONS.  115 

the  measurements  of  its  parts.  This  will  suffice  for  the  frame, 
but  the  motions  of  the  train  must  be  understood,  so  as  to  know 
what  position  to  give  to  other  parts,  corresponding  to  a  given 
position  of  some  one  part. 

4.  In  some  machines,  however,  there  are  subordinate  trains, 
serving  to  adjust  the  position  or  speed  of  the  principal  trains,  as 
in  case  of  engine  governors.  Also  some  parts  are  adjustable 
by  hand,  as  the  position  of  the  bed  in  a  drilling  machine,  or  of 
the  tool  and  rest  containing  it,  in  a  lathe. 

5.  The  working  parts  of  every  machine  consist  of  certain  me- 
chanical elements  or  organs,  which  are  comparatively  few  in 
number,  and  not  always  all  present  in  any  one  machine.  The 
principal  of  these  are  pistons,  cross-heads,  shafts,  cranks,  cams 
and  eccentrics,  toothed  wheels,  screws,  band-pulleys,  connecting- 
rods,  bands  or  chains,  sliding  or  lifting  valves,  grooved  links, 
rocking  arms  and  beams,  flat  or  spiral  springs,  chambered  parts 
and  internal  passages,  as  pump-barrels,  steam-cylinders,  valve- 
chests,  etc. 

6.  These,  considered  separately,  are  of  various  degrees  of  com- 
plexity of  design,  many  of  them  quite  simple.  By  far  the  most 
imj^ortant,  relative  to  the  geometrical  theory  of  their  perfect 
action,  are  toothed  wheels  of  various  forms.  These  we  shall 
therefore  principally  consider,  together  with  a  few  other  useful 
examples. 

Siipporters. 

Example  1.  A  Pillow-block.  Pillow-blocks  of  various 
designs,  adapted  to  horizontal,  or  vertical,  or  beam  engines, 
are  so  common,  and  so  generally  represented  in  works  on  prac- 
tical mechanism,  that  the  following  figure,  taken  from  a  drawing 
to  scale,  is  inserted  here  as  a  sufficient  guide;  the  object,  more- 
over, being  symmetrical  with  respect  to  the  centre  line  00',  and 
a  little  more  than  half  shown. 

Descriptio7i. — BB',  not  definitely  shown  in  plan,  is  a  portion 
of  the  main  bed  of  an  engine.  SS'  is  the  sole  of  the  pillow- 
block,  DD'  its  body,  CO'  its  cover,  and  dd — d'd'  the  brasses 
which  immediately  enclose  the  fly-wheel  shaft  of  the  engine. 
The  holding-down  bolts,  as  b — ¥b',  pass  through  slotted  holes 
pg,  a  little  wider,  that  is,  in  the  direction  pq  than  the  diameter 


SUPPORTERS   AND    CRANK    MOTIONS.  117 

of  the  bolt.  This  construction  allows  for  adjustment  of  the 
position  of  the  block  by  wedges,  driven  between  the  sole  and 
stops  I,  solid  with  the  bed  B'. 

The  cover  CO'  is  held  in  place  by  bolts  c — c'W,  the  heads  h' 
being  in  recesses,  sunk  in  the  under  side  of  the  sole.  The  spaces 
at  r  and  n  between  the  cover  and  the  body,  allow  for  the  wear 
of  the  brasses  dd — d'd'  against  the  shaft.  To  prevent  lateral  or 
rotary  displacement  of  the  brasses,  ears  ee — e'e'  project  from 
them  into  recesses  in  the  cover  and  body  of  the  block.  The  same 
end  is  often  attained  by  making  their  outer  or  convex  surfaces 
octagonal,  and  by  providing  them  with  flanges  where  they  enter 
and  leave  the  block.  00'  is  the  oil  cup,  here  solid  with  the 
cover,  but  oftener  now  a  separate  covered  brass  cup,  contrived 
to  supply  oil  gradually  to  the  shaft. 

Construction. — The  proportions  of  the  figure  being  correct, 
assume  ag,  the  half  length  of  the  sole,  to  be  12  inches,  and  meas- 
ure by  a  scale  its  actual  length  on  the  figure.  A  comparison  of 
the  two  will  indicate  the  corresponding  scale  of  the  figure.* 

Then,  having  determined  the  scale,  all  the  other  measure- 
ments can  be  determined  by  it  to  agree  with  each  other,  and  the 
figure  can  be  draiun  on  any  scale  desired,  from  ^  to  ^  of  the  full 
size. 

The  body  being  symmetrical,  all  the  measurements  to  the  left 
from  00'  can  be  laid  off  to  the  right  of  it,  and  the  complete 
projections  thus  constructed. 

The  method  of  drawing  hexagonal  nuts  has  been  shown  in 
detail  in  Div.  I.,  Problems  31,  32. 

Execution. — Note  the  heavy  or  shade  lines  as  in  previous  ex- 
amples (Div.  XL);  but  if  the  figure  is  to  be  shaded  and  tinted,  ink 
it  wholly  in  pale  lines  or  none. 

Exercises. — 1.  Construct  from  the  two  given  projections  an  end  eleva- 
tiou  of  the  block. 

2.  Construct  a  vertical  section  on  the  centre  line  05. 

3.  Construct  a  top  view  with  the  cover  removed.  (The  dotted  lines 
showing  the  internal  construction  will  enable  these  sectional  views  to  be 
made.) 

*  The  proportions  adopted  by  different  builders,  and  by  the  same  builder 
for  different  cases,  being  not  precisely  alike,  tl^  pupil  is  thus  encouraged 
not  to  think  any  one  set  of  given  measurements  indispensable. 


118  SUPPORTEKS   AXD    CRANK   MOTION'S. 

Ex.  2.     A  Standard  for  a  Lathe.     PI.  XVI.,  Fig.  1. 

Description. — This  example  illustrates  the  application  of  tan- 
gent lines  and  circles  to  the  designing  of  open  frames,  having 
outlines  conveniently  varied  for  use  and  economy  of  material. 

The  figure  shows  half  of  the  side  view  (the  object  being  sym- 
metrical), also  an  edgewise  view.  The  scale,  -J,  being  given, 
and  the  operatioyis  of  construction  being  here  more  important 
than  the  precise  measurements,  only  a  few  of  the  principal 
dimensions  are  given  in  this  and  in  the  following  figures,  leaving 
the  rest  to  be  assumed,  or  sufficiently  determined  by  knowing 
the  scale. 

The  double  lines  on  the  edges  indicate  ribbed  edges,  so  made 
to  secure  stiffness  and  strength.  The  central  and  triangular 
openings  may  also  afford  rests  for  long-handled  tools  or  metal 
bars.     The  nut  n  secures  the  standard  to  the  lathe-bed. 

Construction. — Having  made  the  half  widths  4:^"  and  12^''  at 
top,  and  at  AB,  the  outline  BD  may  be  drawn.  This  is  com- 
posed of  an  arc  of  60°  with  radius  AB,  a  tangent  to  this,  and  a 
second  arc  of  60°  tangent  to  the  last  line,  and  with  its  centre  on 
a  horizontal  line  through  D. 

The  outline  of  the  central  opening  is  partly  concentric  with 
the  arc  through  D,  partly  circular  with  C  as  a  centre,  and 
partly  circular  as  shown  at  the  top,  and  there  tangent  to  the  side 
arcs.  C  is  here  taken  on  a  horizontal  line  through  the  lower 
end  of  the  arc  from  D. 

Execution. — The  figure,  both  halves  of  which  should  be 
drawn,  and  on  a  little  larger  scale,  as  \  or  -J-,  gives  occasion  for 
the  neat  drawing  of  curved  shade  lines,  and  the  neat  connection 
of  tangent  outlines. 

Exercise. — 1.  Vary  the  design  by  making  the  arcs  from  B  and  D  each 
less  than  60°,  and  so  that  C  shall  be  on  the  radius  through  the  lower 
limit  of  the  arc  througli  D. 

Ex.  3.  Section  of  an  Engine-bed,  Guides,  and  their 
Support.     PL  XVI.,  Fig.  2. 

Description. — ABCD  is  a  cross-section  of  the  bed  of  a  hori- 
zontal engine,  which  is  of  uniform  section  throughout,  hh  is  a 
vertical  plate,  bolted i»to  the  bed  as  shown  at  n.  From  this 
plate,  and  solid  with  it,  project  two  or  more  arms  HII,  which 


c 


r^ 


SUPPORTERS   AND    CRAXK   MOTIONS.  119 

support  the  guide-bars  GG,  between  which  slides  the  cross-head, 
not  shown,  to  which  the  outer  end  of  the  piston-rod  is  fastened, 
as  may  be  understood  from  the  equivalent  parts  of  nearly  any 
locomotive  or  stationary  engine. 

Construction. — Only  the  principal  measurements  being  given, 
the  others  can  be  assumed,  or  made  out  by  the  given  scale.  The 
left  side  of  the  bed  having  vertical  faces,  these  may  be  used  as 
lines  of  reference  from  which  to  lay  off  horizontal  measurements. 
Vertical  ones  can  be  laid  off  from  the  base  line  AB;  or,  on  the 
guide  attachments,  from  the  top  of  the  guides  downward.  To 
give  greater  stiffness  to  the  arm  H,  its  lower  principal  curve  is 
struck  from  a  centre,  h,  1^"  to  the  right  of  a,  the  centre  of  its 
semicircular  outlines.  The  curved  outlines  generally  are  com- 
posed of  circular  arcs  tangent  to  each  other. 

As  a  minute  following  of  given  copies  is*  not  intended,  these 
general  explanations,  measurements,  and  scale  Avill  sufficiently 
guide  the  learner  in  the  construction  of  examples  like  the 
present. 

Execution. — The  thin  material  of  the  bed  gives  occasion  for 
section  lines  as  fine  and  close  together  as  can  well  be  made. 

Exercises. — 1.  Reverse  the  figure  right  for  left. 

2.  Supposing  the  guides  to  be  four  feet  long,  make  a  side  and  a  plan 
view,  showing  three  arras  to  the  supporter  H. 

7.  Bearings. — This  is  a  general  term  meaning  any  surface 
which  immediately  supports  a  moving  piece.  The  bearings  of  a 
rotating  piece  are  cylindrical  and  variously  termed.  Journals 
are  formed  in  the  frame  of  a  machine  and  lined  with  brass  or 
other  anti-friction  alloy.  When  detached,  they  are  pillotu-Uochs, 
as  already  shown.  Bushes  are  whole  hollow  cylindrical  linings 
of  journals,  but  being  unadjustable  to  compensate  for  wear, 
separate  brasses  are  better.  Footsteps  are  the  bearings  at  the 
base  of  vertical  shafts,  the  lower  end  of  which  is  a  pivot.  Axle 
boxes  are  the  terminal  supports  of  rail-car  axles,  and  have  a  small 
vertical  range  of  motion  between  the  jaws  of  a  stout  iron  frame. 

Cranhs  and  Eccentrics. 

8.  A  crank,  Fig.  a,  is  an  arm,  CC,  keyed  at  one  end  firmly 
to  a  revolving  shaft   SS'  by  a  key  hh',  and  hence  revolving 


120 


SUPPOKTEES   AKD    CRAXK   MOTIOlirS. 


with  it ;  and  at  the  other  end  carrying  a  cranh-pin pp' ,  which  is 
embraced  loosely  by  a  connecting  rod.  This  connecting  rod 
similarly  embraces  a  parallel  pin  in  the  cross-head  attached  to  a 
piston-rod,  a  pump-rod,  or  other  piece  having  a  reciprocating 
motion.  Thus  a  rectilinear  recijorocating  motion,  as  of  a  piston, 
is  conyerted  into  a  rotary  motion,  as  seen  in  any  locomotire,  or 


Ficj.  a. 

a  stationary  engine  of  the  iisual  type,  or  vice  versa,  as  in  case  of 
a  pump.  The  length  of  the  strolce  of  the  piston  must  evidently 
be  equal  to  the  diameter  of  the  circle  described  by  the  centre  of 
the  crank-pin ;  that  is,  equal  to  twice  Sp.  See  Fig.  h,  where 
the  stroke  qq^  of  the  forward  end  of  the  connecting  rod  mr  is 
equal  to  pp^,  the  dotted  circle  being  that  described  by  the  centre 
of  the  crank-pin. 

9.  Fig.  l  illustrates  an  important  elementary  point  in  crank 
motions.     Remembering  that  any  connecting  rod  is  of  invariable 


length,  take  the  middle-point  m-  of  the  stroke  qq^^%  a  centre,  and 
tlio  length  ???S  =z  qp  =^  q^p^  of  the  rod  as  a  radius,  and  the  arc 


SUPPORTEES   Al^J)    CRAXK   MOTIOXS. 


121 


rSri  thias  described  will  intersect  the  crank-pin  circle  in  the 
corresponding  positions  r  and  r^  of  the  crank-pin. 

Thns,  while  the  cross-head  j^in  passes  over  mq  and  qm,  the 
crank-pin  describes  the  arc  Tipr,  greater  than  a  semicircle  ;  but 
while  the  former  is  passing  over  mqi  and  qiin,  the  crank-pin 
proceeds  over  rp^r^  less  than  a  semicircle. 

Conversely,  while  the  crank-pin  traverses  the  rear  semicircle 
A/jB,  the  cross-head  jjin  only  travels  from  n  to  q  and  back  ;  but 
when  the  crank-pin  describes  the  semicircle  BjOiA,  the  other 
pin  travels  from  oi  to  q^  and  back ;  An  being  equal  to  Vim. 

With  the  use  of  the  connecting  rod,  this  inequality  mn,  be- 
tween the  two  partial  double  strokes  (^nq  and  Qw^-i,  would  dis- 
appear only  by  using  a  rod  of  infinite  length.  But  the  em- 
ployment of  a  yoke  with  a  slot,  equal  and  parallel  to  AB,  as  in 


C 


-} 


Fig.  c. 

Fig.  c,  produces  the  same  result  by  finite  means.  Here  a  piston- 
rod,  P,  issuing  from  the  steam-cylinder  C,  is  rigidly  attached  to 
a  yoke,  AB,  in  which  the  crank-pin 
plays  as  it  is  driven  by  the  yoke.  In  this 
case  the  piston  is  exactly  at  the  middle 
point  of  its  stroke  when  the  crank-pin 
is  at  either  end  of  the  diameter  AB. 
This  movement  is  often  seen  in  steam 
fire-engines. 

10.  Eccentrics. — The  distance,  Fig.  a, 
from  the  centre  of  tlie  shaft  S  to  the 
centre  of  the  crank-pin  p,  is  called  the 
a7'tn  of  the  crank.  When  this  arm  is  so  short,  as  comj)ared  with 
the  diameter  of  the  shaft,  as  to  be  entirely  within  the  shaft,  as 
at  S^,  Fig.  cl,  the  crank-pin  AB,  whose  centre  is  p,  has  to  be 


122 


SUPPORTERS    AND    CRANK   MOTIONS. 


made  large  enough  to  embrace  the  shaft.  In  this  case  the  crank- 
pin  is  called  an  eccentric. 

That  the  eccentric  is  simply  a  short  crank  in  principle  and 
action,  will  be  evident  by  substituting  for  the  crank  C,  Fig.  a, 
a  circular  plate  with  centre  p  and  radius  sufficient  to  include 
the  shaft.  In  either  form  of  Fig.  a,  and  in  Fig.  d,  a  connecting 
rod  attached  to  the  crank-pin  would  actuate  any  piece  at  its 
opposite  end  through  a  stroke  equal  to  twice  S/). 

11.  A  connecting  rod  is  attached  to  a  crank-pin  by  a  method 
having  many  modifications  in  minor  details.  The  general  prin- 
ciple, alike  for  all,  is  shown  in  Fig.  e.  The  object  to  be  secured 
is  an  invariable  distance  between  the  centres  of  the  crank-pin 
and  the  pin  p,  at  which  the  rod  R  is  attached  to  the  cross-head. 
E  IS  the  end  of  this  rod,  called  the  stub-end.     ssss  is  the  strap. 


'b      'b 


^  Fig.e. 


in  one  piece.  B,  shown  sectionally,  and  B'  in  elevation,  are  the 
brasses,  square  outside  and  cyhndrical  inside,  which,  together, 
embrace  the  shank  of  the  crank-pin,  and  are  kept  from  sliding 
off  by  the  head  of  the  crank-phi  h,  Fig.  a.  The  whole  is  fas- 
tened by  two  bolts  bb,  bb. 

This  arrangement  being  understood,  sup^iose  that  by  long 
wear  the  brasses  play  loosely  upon  the  pin  p.  By  driving  in  the 
slightly  tapering  key  ^•^•,  its  side  aa  presses  the  brass  B  against 
the  pin  p,  the  width  of  the  slots  ac  ac  in  the  strap  permitting 
this  to  be  done.  Then  loosening  the  bolts  b,  the  holes  for  which 
are  oblong  from  right  to  left,  as  seen  in  a  plan  view,  a  further 
driving  in  of  the  kcj  kic  operates  through  the  hooked  piece  ^, 
called  a  gib,  to  draw  the  strap  s  to  the  right,  and  thus  draw  up 
the  brass  B'  against  the  pin  p. 

Having  understood  one  construction,  the  learner  will  be  able 


SUPPORTERS   AND    CRANK    MOTIONS. 


123 


to  understand  all  the  modifications  which  he  may  notice  on  loco- 
motive or  other  engines,  such  as  the  omission  of  the  gib,  which 
is  unnecessary,  with  the  bolts ;  a  separate  key  for  each  brass ; 
the  stub-end  extending  to  the  left  of  kh,  so  as  to  wholly  enclose 
it,  when  no  bolts  would  be  necessary ;  a  screw  motion  at  the 
small  end  of  h  for  drawing,  instead  of  hammering  in  the  key;  etc. 

Ex.  4.  A  Crank.  PL  XVI.,  Fig.  3.  This  consists  essen- 
tially of  two  collars  connected  by  a  tapering  arm,  the  whole  in 
one  cast-iron  piece.  0  is  the  centre  of  the  8-inch  shaft,  and 
shaft  collar  of  diameter  ad,  19''.  P  is  the  centre  of  the  crank- 
pin,  of  A!',  and  of  its  collar,  of  9"  diameter.  The  arm  CO'  is 
chambered  as  indicated  by  the  dotted  line  around  C.  The 
linear  arm  OP  is  24". 

The  surfaces  of  the  arm  flow  into  those  of  the  collars  as  indi- 
cated in  line  drawings  by 
the  curved  ends  of  the 
upper  edge  of  0',  lines 
whose  geometrical  con- 
struction is  unnecessary  in 
practice,  but  may  be  found 
as  follows.  Fig.  /.  In  this 
figure  the  arc  c'n'  is  that 
at  en,  Plate  XVI.,  Fig.  3, 
enlarged,  and  the  line  On 
corresponds  to  PO.  Then 
project  points  of  c'n'  upon 
en,  as  r'  at  r,  and  r7\  and 
r'r"  are  the  tAvo  projec- 
tions of  the  horizontal 
circle  through  rr',  which 
cuts  the  edge  7i-iP  of  the 
crank  at  Vi,  which,  pro- 
jected upon  the  horizontal 
line  r'r",  srives  r".    Simi- 


larly, other  points  of  the 


Fig.f 

required  curve  n'li"  r"o'  are  found.  Such  curves  are,  however, 
after  the  full-sized  construction  of  a  few  cases,  to  apprehend  their 
general  form,  sketched  by  hand,  as  they  are  not  essential  to  a 
working  drawing. 


124:  SUPPORTERS   AND   CRAN'K   MOTIOKS. 

Exercises — 1.  Complete  the  crank,  half  of  •u-liicli  is  shown  in  the  figure. 

2.  Make  a  longitudinal  section  of  the  crank. 

3.  Draw  from  measurement  any  accessible  ribbed,  trussed,  or  cham- 
bered crank. 

4.  Draw  a  cranked  axle  (such  as  may  be  seen  on  old  locomotives  hav- 
ing "inside  connections"). 

Ex.  5.     A  Ribbed  Eccentric  and  Strap.     PI.  XVI., 

Figs.  4,  5.  This  may  also  be  called  an  open  or  skeleton  eccentric. 
0  is  the  centre,  and  Oh  the  radius  of  the  shaft,  S\"  diameter, 
to  which  the  eccentric  is  clamped  by  clamp-screws,  one  of  which 
is  n.  The  centre  of  the  eccentric  is  a,  which  makes  the  crank- 
arm  0«  of  the  eccentric  d",  and  hence  the  stroke,  called  the 
throw,  of  the  valve,  or  whatever  piece  is  moved  by  the  eccen- 
tric, Q". 

The  width  of  the  different  parts  of  the  eccentric  is  shown  on 
the  fragment  of  sectional  view,  as  at  o'o"  the  thickness  of  the 
flange,  or  feather,  o,  the  width  e'e"  of  the  collar  h  and  rim  e, 
and  the  width  of  the  rib  c. 

Fig.  5  shows  a  little  more  than  one-quarter  of  the  strap  which 
surrounds  the  eccentric,  and  a  little  more  than  half  of  one  of  its 
two  halves,  which  are  bolted  together  tlirough  the  ears,  as  cd. 
The  arc  ab  is  of  the  radius  ae,  Fig.  4 ;  mn  is  of  the  radius  ac, 
thus  showing  the  groove  in  the  strap  wliich  just  fits  the  rib  on 
the  eccentric,  and  so  i)rcvents  the  strap  from  slipping  off.  The 
portion  of  the  strap  shown  carries  the  socket  ne,  in  which  is 
keyed  or  clamped  the  eccentric  rod,  corresponding  to  the  con- 
necting rod  of  a  crank.  The  opposite  or  left-hand  half  is  unin- 
terrupted in  outline.  The  figure  r'g'  shows  the  form  of  the 
section  at  rg. 

Execution. — The  numerous  tangent  arcs  and  curved  heavy 
lines  tapering  at  their  termination  will  afford  occasion  for  special 
care. 

Exercises. — 1.  Draw  the  whole  of  the  eccentric  and  its  strap. 

2.  Make  a  horizontal  section  of  the  eccentric. 

3.  Make  an  end  elevation  of  the  eccentric  and  strap. 

Ex.  6.  A  Grooved  Eccentric.  PL  XVI.,  Figs.  6,  7. 
Tliis  might  also  ])C  called,  by  reason  of  its  form,  a  chambered  or 
box  eccentric,  since  all  of  it  between  the  solid  collar  Qa  and  the 


PL  XVI 


r 


SUPPOKTERS  AND   CRANK   MOTIONS.  125 

rim  cd  consists  essentially  of  two  thin  plates  enclosing  a  hollow 
interior  of  width,  '6%" ,  shown  on  the  fragment  of  end  elevation 
— the  scale  is  jV?  ^^  in  Ex.  5. 

The  shaft  opening,  of  centre  0,  is  G"  in  diameter,  surrounded 
by  solid  metal  \"  thick  as  indicated.  The  arm  OQ  being  ^:\" , 
makes  the  throw  of  this  eccentric  8|-".  The  opening,  P,  5" 
diameter  in  the  walls  of  the  eccentric,  gives  access  to  the  clamp- 
screw  n  by  which  it  is  fastened  to  the  shaft. 

The  strap.  Fig.  7,  a  section  of  Avhich  is  shown  at  H,  sets  in 
the  groove  c'c"  of  the  circumference  of  the  eccentric.  Its  outer 
arc  mn  is  drawn  from  a  centre  a  little  to  the  right  of  that  of  AB, 
so  as  to  support  the  rod-socket  BC.  As  in  the  last  example,  the 
strap  is  in  two  halves  bolted  together  through  ears  as  at  D. 

Exercises. — 1,  Draw  the  whole  eccentric  with  the  strap  in  place  upon 
it,  and  an  end  view  of  both. 

2.  Make  a  horizontal  and  a  vertical  section  (perpendicular  to  the 
paper)  of  the  combined  eccentric  and  strap. 

12.  Chech,  loch,  or  janih  nuts. — On  parts  of  machinery  which 
are  exposed  to  a  jarring  motion  at  high  speed,  as  in  locomotive 
machinery,  two  nuts  are  commonly  seen  at  the 
end  of  the  bolts  which  secure  such  pieces.  These 
serve  to  clamp  each  other  against  the  screw- 
threads  of  the  bolt,  and  thus  hold  each  other 
from  working  off  the  bolt. 

Other  contrivances  for  securing  the  same  re- 
sult, are  nuts  with  notched  sides,  into  which  a 
detent  enters,  as  may  be  observed  in  winding  up  ^'^9-  9' 

a  watch;  or  a  forked  key  hh'  through  the  bolt  and  outside  of 
the  nut,  as  in  Fig.  q. 


fc 


CHAPTER  II. 

GEARING. 

13.  Gearing  is  the  term  applied  to  wheels  or  straight  bars 
when  they  are  armed  with  interlocking  teeth  enabling  them  to 
take  a  firmer  hold  of  each  other,  for  the  jiurpose  of  communi- 
cating motion,  than  they  could  if  they  were  smooth  surfaces, 
tangent  to  each  other  and  communicating  motion  only  by  means 
of  the  friction  of  their  surfaces  of  contact. 

In  order  to  a  smooth  and  uniform  motion,  the  teeth  must  be 
equal  and  equidistant,  and  those  on  each  body  adapted  to  the 
form  of  those  on  the  other  body.  Also,  in  order  that  the  toothed 
bodies,  two  cylinders  tangent  to  each  other,  for  exami)le,  should 
preserve  the  distance  between  their  centres,  depressions  heloto 
the  original  surface  of  each  body  must  be  made  between  the  teeth, 
in  order  to  receive  the  portions  which  project  teyond  that  of  the 
other  body. 

A  toothed  bar  is  called  a  raclc;  a  toothed  cylinder,  a  spur-ivlieel, 
or  pinion  if  small ;  a  toothed  cone,  a  conical  or  level  wheel. 

14.  Forms  of  teeth.— 1°.  When  a  circle  C,  PI.  XVII.,  Fig.  1, 
rolls,  without  slipjjing,  on  a  fixed  straight  line  AB,  any  one  i^oint 
of  the  circle  describes  the  curve  called  a  cycloid.  Thus  the  point 
0  describes  the  cycloid,  one-half  of  which  is  OE',  while  the  circle 
C  rolls  to  the  position  C. 

2°.  When,  on  the  contrary,  a  straight  line  as  3,3',  Fig.  2,  rolls, 
without  slipping,  on  a  fixed  circle,  any  point  of  the  rolling  line 
describes  the  curve  called  an  involute  of  the  circle.  Thus  3'  de- 
scribes the  involute  3',  2',  0. 

3°.  Again:  when  one  circle  rolls  on  the  exterior  of  another  as 
the  circle  BB',  Fig.  3,  on  the  circumference  BFA,  any  point  on 
the  rolling  circumference  traces  the  curve  called  an  epicycloid. 
Thus  the  point  P  traces  the  half  epicycloid  EG,  while  the  circle 
C  rolls  to  the  position  AG. 

4°.  Finally:  when  BB',  instead  of  rolling  on  the  convex  or  ex- 
terior side  of  the  circumference  C,  rolls  upon  its  concave  side. 


GEAEIXQ.  127 

or  within  it,  any  point  of  the  rolling  circle  generates  the  curve 
called  a  hypocycloid. 

In  all  these  cases  the  fixed  line  is  called  the  base  line  or  circle. 

15.  Suppose  now  the  circle  BB',  Fig.  3,  to  be  revolved  180° 
about  a  tangent  at  B.  It  would  then  be  tangent  to  the  circle  C 
interiorly  at  B,  as  it  now  is  exteriorly  at  that  point. 

If  then,  the  diameter  BB'  being  less  than  radius  CB,  the  circle 
BB'  rolls  within  C,  on  the  arc  BFA,  the  hypocycloid  traced  by 
B  will  be  alove  AB.  But  if  a  circle  of  diameter  greater  than  CB 
were  to  roll  within  C  on  the  arc  BFA,  the  hypoc^'Cloid  curve 
would  be  heloiu  AB.  It  plainly  follows  that  the  hypocycloid 
traced  by  the  point  B,  when  the  circle  of  diameter  just  equal  to 
CB  rolls  within  C  on  the  arc  BFA,  would  coincide  with  BO. 
That  is:  the  hypocycloid  traced  hy  any  point  of  a  circumference 
ivliich  rolls  on  the  inside  of  a  base  circle  of  twice  its  diameter, 
,  is  a  straight  line. 

16.  These  four  curves  (14)  are  suitable  forms  for  the  teeth  of 
wheels,  for  the  simple  reason  that  when  two  circles  as  C  and  C, 
PI.  XVII. ,  Fig.  4,  maintain  rolling  contact  with  each  other,  as 
at  H,  equal  arcs  of  each  come  in  contact  in  a  given  time.  Hence 
the  motion  is  the  same  in  effect  as  if,  separately,  C  rolled  on  C 
as  a  fixed  circle,  and  then  C  rolled  over  an  equal  arc  on  C  as  a 
fixed  circle,  and  therefore  contact  Avill  be  maintained  by  arming 
the  wheels  with  teeth  generated  by  the  point  H,  for  each  case 
respectively. 

Similarly  for  a  wheel  C,  and  rack  I'J'. 

The  circle  or  line  employed  for  generating  the  tooth  curves  is 
called  their  generating  circle,  or  line. 

17.  Construction  of  tooth  curves. — This  is  very  simple,  and 
follows  directly  from  the  definitions  in  (14).  Cycloid.  Thus  PI. 
XVII.,  Fig.  1,  the  points  1,  2,  3,  etc.,  on  the  circle  C  indicate 
the  heights  of  0  above  AB,  corresponding  to  1,  2,  3,  etc.,  on  AB 
as  successive  points  of  contact  of  C  with  AB.  Then  the  inter- 
sections of  the  parallels  to  AB  through  1,  2,  3,  etc.,  on  C,  with 
the  arcs  of  radii  al,  h2,  c3,  etc.,  will  be  points  of  the  cycloid  OE'. 
Both  sets  of  spaces  01, 12,  etc.,  are  equal,  since  there  is  no  slipping 
of  the  circle  in  rolling  on  AB.  Epicycloid.  Likewise  in  Fig.  3, 
arcs  Fl,  etc.,  on  circle  C,  =  arc  Fl,  etc.,  on  circle  C^  express 


128  QEARIN-Q. 

the  character  of  the  motion;  a,  h,  c,  etc.,  are  positions  of  the 
centre  C  corresponding  to  5,  4,  3,  etc.,  as  points  of  contact  of 
the  circles;  the  arcs  of  radii  Cl,  C3,  C3,  show  the  radial  dis- 
tances of  F  from  BFA  as  C  rolls  on  C;  hence,  finally,  the  iinter- 
sections,  not  lettered,  of  the  arcs  of  radii  e\  and  Cl,  d'Z  and  C2, 
c3  and  C3  are  points  of  the  epicycloids  FG  and  FE'. 

Tlie  hypocycloid  is  constructed  in  a  precisely  similar  manner. 

The  involute  is  approximately  rej)rcsented  by  tangent  circular 
arcs  as  in  Fig.  2.  Here,  11',  22',  33',  being  positions  of  the  roll- 
ing straight  line  at  equidistant  points  of  contact  1,  2,  3,  we 
describe  the  arc  01'  with  radius  10  (taking  the  chord  us  approxi- 
mately equal  to  the  arc),  then  an  arc  1'  2'  with  radius  21',  then 
the  arc  2'  3'  with  radius  32',  etc.  The  more  numerous  the 
points  1,  2,  3,  etc.,  the  closer  Avill  the  compound  curve  thus 
found  approximate  to  a  true  involute. 

These  curves  can  le  constructed  mechanically  on  a  large  scale 
by  means  of  a  pin  or  pencil  point  inserted  firmly  in  the  edge  of 
a  wooden  ruler  or  circle,  cither  of  wliicli  is  made  to  roll  without 
slipping  on  the  other,  or  the  circle  on  a  fixed  circle;  or  within  a 
circular  oj^cning  in  a  thin  board,  in  the  case  of  the  hyi^ocycloid. 

18.  Definitions. — Let  the  circles  of  radii  CH  and  C'H,  PI. 
XA^IL,  Fig.  4,  represent  the  original  circumferences  of  two 
cylinders  having  IJ  for  a  common  tangent  at  H,  and  now  pro- 
vided Avith  interlocking  teeth  as  shown,  forming  a  pair  of  sp7ir- 
tvheels.  These  circles  are  called  pitch-circles.  The  correspond- 
ing line  I'J'  of  the  racJc  is  called  its  pitch-line. 

The  distance  ab,  or  II'K  on  the  rack,  which  includes  a  tooth 
and  a  space  on  the  pitch-circle,  is  called  the  pitch. 

The  circle  of  radius  Cc  is  t\ic  root-circle,  and  contains  i\\Q7'oots 
of  the  teeth. 

The  circle  of  radius  Cd  is  the  ])oi)it-circle,  and  coiitains  the 
points  of  tlic  teeth. 

The  sui'faccs  as  hd  are  the  faces  of  the  teeth,  and  those  as  be 
are  ihc'w  flafiks. 

10.  Usual  proportions. — Supposing  the  pitch  divided  into  15 
equal  parts,  7  of  these  are  taken  for  the  width,  ah,  of  the  tooth, 
leaving  8  of  them  for  the  width  of  the  space,  hb,  to  allow  easy 
working  of  the  teeth.     Also  5 J  of  these  spaces  are  taken  for  the 


GEAUIXG.  129 

radial  extent  of  the  teeth  beyond  the  pitch-circle  and  (>\  of  them 
for  their  depth  below  the  pitch-circle,  to  prevent  the  tootli  points 
of  one  wheel  from  striking  the  rim  of  the  other  wheel. 

Application  of  Tooth  Curves. 

20.  Designing  of  gearing. — Comparing  (15)  and  (IG),  the  gen- 
erating circle  of  the  tooth  curves  must  be  smaller  than  the  pitch- 
circle  in  order  to  form  the  necessary  flank  surfaces  (18).  A 
common  practice  is,  to  employ  for  the  flanks  of  each  wheel  a 
generating  circle  of  diameter  equal  to  the  radius  of  the  i^itch-cir- 
cle  of  that  wheel;  in  order  to  produce  radial  Hanks,  as  most 
simple.  Now,  as  seen  by  inspection  of  PI.  XVIL,  Fig.  4,  the 
face  of  a  tooth  of  each  wheel  is  in  contact  with  ihcjlanh  of  some 
tooth  of  the  other  wheel.  Hence  (IG)  the  same  circle  that  gen- 
erates the  flanhs  of  one  wheel  must  generate  the  faces  of  the 
teeth  of  the  other,  since  keeping  the  generating  circle  of  diameter 
CH  in  contact  with  the  pitch-circles  at  their  point  of  contact  II, 
requires  in  effect  the  equal  rolling  of  that  generating  circle  upon 
the  exterior  of  the  circumference  of  wheel  C,  and  on  the  interior 
side  of  that  wheel  C.  This  can  easily  be  seen  experimentally  by 
using  three  card-board  circles. 

21.  Detailed  description. — Ep icy cloided  teeth. — The  circle  CH 
generates  the  radial  flanks,  as  be,  of  the  teeth  of  C  (14,  4°),  and 
by  rolling  on  the  exterior  of  C  generates  tlie  face  curves  of  the 
teeth  of  C.  To  avoid  confusion,  HK  may  represent  one  of  these 
curves,  though  it  is  really  an  involute.  Likewise,  the  circle  CH 
generates  the  radial  flanks  of  wheel  C,  and  by  rolling  on  the 
exterior  of  C  will  generate  the  epicycloidal  faces  of  its  teeth, 
found  as  in  Fig.  3,  but  represented  as  before  by  the  involute  IIL. 

22.  Objections. — Each  wheel  having  a  separate  generating 
circle,  each  will  work  correctly  only  with  the  other.  But  if  one 
uniform  generating  circle  be  employed  for  the  faces  and  Hanks 
of  any  number  of  different-sized  wheels  of  the  same  pitch,  any 
two  of  them  will  work  together  properly.  This  common  gen- 
erator must  not  exceed  half  the  size  of  the  least  wheel  of  the  set, 
so  as  to  avoid  convex  flanks  (15). 

23.  Involute  i'rr'/Z(.— Involute  faces  for  both   wheels   can    be 


130  GEARING. 

formed  as  shown  by  the  rolling  of  the  common  tangent  IJ  at  H, 
first  on  C,  giving  the  involute  face  curve  HL  (Fig,  4),  and  then 
on  C,  giving  the  face  curve  HK. 

Usually,  however,  when  involute  teeth  are  employed,  they  are 
not  combined  with  radial  flanks,  since  this  violates  the  principle 
that  the  same  generatrix  should  form  the  face  and  the  flank 
which  are  to  be  m  contact ;  but  IJ  is  made  a  common  tangent 
through  11  to  the  root-circles  of  the  two  wheels,  so  that  the  in- 
volute teeth  will  be  bounded  by  a  single  involute  curve  reaching 
to  the  root-circles,  as  they  should,  since  a  straight  line  cannot 
be  rolled  on  the  interior  side  of  the  pitch-circles  to  i)roduce  sepa- 
rate flank  curves. 

24.  The  rach-generating  circle  continuhig  to  be  half  the  size 
of  its  own  pitch-circle,  the  generating  circle  for  the  rack  flanks 
will  be  I'J',  since  a  straight  line  is  a  circle  of  infinite  radius  and 
half  of  that  radius  is  infinite  still.  This  understood,  the^^aw^-s 
of  the  rack  are  straight  lines  as  AH'  perpendicular  to  its  pitch- 
line,  and  \X\Q  facea  of  the  teeth  of  C,  being  properly  generated 
by  the  same  line,  are  involutes  as  H'F'. 

Likewise  \X\(iflanhs  of  the  teeth  of  C  are  straight  lines  genera- 
ted by  the  circle  of  diameter  ClI',  while  the  faces  of  the  rack 
teeth  are  cycloids  as  H'G'  generated,  as  shown,  by  the  rolling 
of  the  same  circle  on  the  pitch-line  I'J'.  But,  as  before,  one 
generating  circle  can  be  used  for  faces  and  flanks  of  both 
wheels. 

Ex.  7.    The  Drawing  of  a  Spur-vrheel. — It  is  convenient 

that   the   pitch   should    be   some   simple   measure  as   1",   1^", 

li",  .  .  .  V,  .  .  .  2^-",  .  .  .  etc.,  and   it   is   necessary   that 

the  pitch  should  be  contained  an  exact  number  of  times  in  the 

])itch-circle.     Hence  the  usual  jn-oblcm  is:    Given  the  pitch  and 

number  of  teeth  of  a  pair  of  wheels,  to  find  their  radii. 

Let  P  =  ])itch  X,  =  number  of  teeth,  R  =  radius, 

and  C  =  circumference  of  pitch-circle. 

Then  C  =  P  X  N  =  3.141G  X  2R, 

P  X  N 
whence  11  = i^ir'  °^  denoting  as  usual,  3.141G  by  it 


R  = 


P  X  N 

2  n    ' 


GEAKING.  131 

Suppose  a  wheel  of  34  teeth  and  IV'  pitch.  Its  circumference 
will  thus  be  30"  and  its  radius  very  nearly  52". 

The  four  quarters  of  the  wheel  being  alike,  it  is  sufficient  to 
draw  one  of  them,  with  the  lirst  tooth  on  the  adjacent  quarters, 
and  this  can  conveniently  be  done  on  a  scale  of  half  the  full 
size. 

Three  forms  of  Avheels  are  in  use  according  to  their  size:  solid 
wheels,  as  in  PL  XVII.,  Fig.  4;  plale  wheels,  consisting  of  a  cen- 
tral hub,  or  boss,  keyed  to  the  shaft,  and  connected  by  a  thin 
plaie  to  the  rim  which  carries  the  teeth;  and  armed  Avheels,  in 
Avhich  the  boss  is  connected  with  the  7'im  by  arms  the  perpen- 
dicular section  of  which  is  often  an  equally  four-armed  cross. 

Attendiug  at  first  principally  to  the  teeth,  let  the  wheel  now 
drawn  be  solid. 

Divide  a  quadrant  of  tlie  pitch-circle  carefully  into  six  equal 
parts,  one  of  which  will  be  the  pitch. 

Proportion  the  teeth  by  (19),  giving  the  root  and  point  circles. 
Lay  off  half  the  width  of  a  tooth  oa  each  side  of  each  point  of 
division  of  the  pitch-circle,  which  will  make  the  lines  as  CH' 
and  CD,  Fig.  4,  centre  lines  of  teeth  instead  of  as  shown  in 
that  figure. 

While  teeth  are  shown  in  detailed  working  drawings,  of  full 
size  and  by  the  most  accurate  construction  of  their  proper  forms, 
they  are  approximately  represented  in  general  illustrative  draw- 
ings, by  various  simple  methods.  Thus  the  faces  may  well  be 
drawn  by  taking  the  pitch  ab  as  a  radius,  with  the  centre,  as  at 
b,  on  the  pitch-circle  to  draw  the  face  aJ).  A  more  summary 
process  is  shown  in  PI.  XIX.,  Fig.  6.  The  flanks,  if  not  radial, 
as  shown  in  the  figure,  should  be  in  reality  hypocycloids  (14,  15) 
which  would  diverge  totoards  the  centre  C,  and  which  may  suffi- 
ciently be  represented  by  taking  d,  for  example,  for  the  centre  of 
the  flank  beginning  at  n. 

Thus  the  elevation  may  be  completed,  placing  the  shade,  or 
heavy  lines,  on  each  tooth  by  the  usual  rule,  as  shown. 

For  the  plan,  draw  two  parallel  lines  at  a  distance  apart  equal 
to  the  luidth  of  the  Avlieel;  that  is,  the  length  of  the  teeth,  Avhich 
may  be  twice  the  pitch.  Then  simply  project  down  the  point 
angles  as  d,  and  visible  root  angles  as  c,  and  the  points  of  con- 
tact of  the  face  curves  with  tangents  parallel  to  IJ,  as  at  a  and 


132  GEARIXG. 

near  h.     To  become  familiar  with  the  subject,  work  out  fully 
the  following  : 

Exercises. — 1.  Construct  the  half,  not  shown,  of  tlie  cycloid.  PI. 
XVII.,  Fig.  1. 

2.  Complete  both  of  the  epicycloids  half-shown  in  Fig.  ?>,  one  with  the 
diameter  of  C  eijuul  to  the  radius  of  C.  Also  one,  given  by  making  the 
circles  C  and  C  equal. 

3.  Construct  the  liypocycloid  generated  by  the  jioint  II  of  circle  C'll, 
Fig.  3,  in  rolling  within  circle  C. 

4.  Construct  the  liypocycloid  generated  by  the  circle  CII'  rolling 
within  tlic  small  circle  C. 

5.  Construct  an  arc  of  the  involute  of  circle  C,  generated  by  the  point 
H'  of  the  line  I'J',  and  by  dividing  a  quadrant  of  C  into  eight  equal 
parts. 

G.  Draw  a  spur-wheel  and  rack,  the  wheel  having  33  teeth  and  2" 
pitch.  Make  the  drawing  of  full  size,  showing  a  quadrant  only  of  the 
wheel,  bisected  at  its  point  of  contact  with  the  rack,  and  let  the  faces 
and  flanks  of  both  pieces  have  one  generating  circle  whose  diameter  shall 
be  -J  tiiat  of  the  radius  of  the  wheel. 

7.  In  Ex.  G,  substitute  for  the  rack  a  wheel  of  20  teeth,  and  let  the 
common  generating  circle  of  the  teeth-profiles  of  both  wheels  be  of  less 
diameter  than  the  radius  of  the  smaller  wheel. 

8.  Draw  enough  of  a  four-armed  wheel  of  30  teeth  and  1\"  pitch  to 
show  two  arms  fully,  making  the  tliickuess  of  the  rim^  and  of  the  arms, 
and  of  the  feather,  and  their  width  also  (see  o,  PI.  XVI.,  Fig.  4), 
which  surrounds  the  openings  between  the  arms,  all  equal  -^'^  of  the 
pitch,  and  the  radial  thickness  of  the  hub  ^  of  the  pitch. 

9.  Draw  PI.  XIX.,  Fig.  G,  twice  its  present  size  or  larger,  and  first 
with  involute  teeth,  and  then  with  epicycloidal  faces  and  hypocycloidal 
flanks,  and  after  constructing  carefully  one  tooth-profile,  find  by  trial 
the  centre  and  radius  of  the  circular  arc  which  will  most  nearly  coincide 
with  it,  to  use  in  drawing  the  other  teeth.' 

25.  Velocities. — It  is  clear  (13)  that  if  one  wheel  has  30  teeth 
and  another  GO,  the  former  must  make  two  revolutions  to  one  of 
the  latter,  also  that  the  radius  of  the  former  is  one  half  that  of 
the  latter.  What  is  true  for  one  such  case  is  evidently  true  in 
principle  for  all  cases.  That  is,  the  number  of  revolutions  in  a 
given  time  of  each  of  a  i)air  of  toothed  Avheels  is  inversely  as  its 
number  of  teeth,  or  as  its  radius. 

The  rircumference  velocities,  as  at  tlie  point  of  contact  II, 
PI.  A'Vir.,  y\'y.  4,  are  necessarilv  e(|u;il.  luit  the  velocities  at  the 


GEARING.  133 

same  distance  from  the  centre,  as  1  foot,  on  both  wlicels  are  as 
the  numbers  of  revolutions,  and  hence  inversely  as  their  radii. 
The  latter  are  termed  angular  velocities.  Then  denoting  them 
by  V  and  v  for  the  wheels  C  and  C  respectively,  and  the  radii 
CII  by  11  and  C'H  by  r,  we  have 

■    V:v::r:Pt. 

Bevel  and  Mitre  WJieels. 

26.  PI.  XVIIL,  Fig.  1,  shows  a  pair  of  level  wheels.  These 
consist  of  a  pair  of  frusta  of  cones,  CAD  and  one  of  which  CAB 
IS  the  half,  provided  with  teeth  which  converge  to  the  common 
vertex,  0,  of  the  cones,  whose  axes,  CB  and  CF,  may  make  any 
angle  with  each  other. 

When,  as  m  Fig.  2,  the  axes  are  at  right  angles,  the  wheels 
are  distinguished  as  mitre  wheels. 

As  C  IS  lowered  nearer  and  nearer  to  AB,  still  continuing  the 
common  vertex  of  the  cones,  the  wheel  AB  becomes  flatter  and 
flatter,  and  when  finally  C  passes  below  AB,  the  wheel  AB  be- 
comes a  hollow  frustum  toothed  on  its  inner  surface. 

On  account  of  the  intersection  of  the  axes  of  bevel  wheels, 
one  or  both  of  the  axes  terminate  at  the  wheels,  as  m  Figs.  1 
and  2. 

27.  Velocities. — The  principles  of  (25)  apply  to  bevel  wheels. 
Hence  having  given  one  wheel,  as  CAB,  Fig.  1,  and  the  ratio 
of  the  velocities,  make  a'd'  and  a'e'  in  this  ratio,  a'd'  repre- 
senting the  relative  velocity  of  the  required  wheel,  and  Ce'  will 
be  the  axis  tangent  from  C  to  an  arc  of  centre  a'  and  radius  a'e', 
and  AD  the  diameter  of  th(j  latter  wheel. 

Or,  having  given  the  vertex  C,  and  axes  CB  and  CF,  set  off 
Cc'  and  C5'  inversely  as  the  two  velocities  (that  is,  set  off  on 
each  axis  a  distance  proportional  to  the  velocity  of  the  other 
axis),  and  complete  the  parallelogram  Cl'a'c',  and  CA  is  the 
line  which  will  divide  the  angle  BCF  included  by  the  axes,  so 
as  to  give  the  radii  AB  and  iAD  of  the  required  wheels. 

When,  as  in  Fig.  2,  the  axes  are  at  right  angles,  the  latter 
construction  applies,  but  the  parallelogram  becomes  the  rect- 
angle CJaO. 


134  GEARING. 

Ex.  8.    To  Draw  a  Pair  of  Bevel  Wheels.    PL  XVIIL, 

Figs.  2-5. 

Let  the  cones  CAB  and  CAD — called  the  pitch-cones,  because 
they  contain  the  pitch-circles — be  given.  At  A,  the  point  of 
contact  of  the  pitch-circles,  draw  EAF  perpendicular  to  CA,  and 
draw  EA,  EB,  FA,  FD.  Then  EAB  and  FAD  will  be  the 
cones  containing  the  larger,  or  outer  ends  of  the  teeth.  Next, 
laying  off  Al  equal  to  the  length  of  a  tooth,  and  drawing  IR  par- 
allel to  AD,  IH  parallel  to  AB,  and  GIJ  parallel  to  EF,  we  have 
JIR  and  GIH,  the  cones  containing  the  inner  ends  of  the  teeth. 

The  wheels  here  shown  have  respectively  36  and  28  teeth. 
Then  divide  each  quadrant  of  the  semi-pitch-circle  on  A'B'  into 
9  nine  equal  parts,  and  each  quadrant  of  the  semi-pitch-circle  on 
A"D'  into  7  equal  parts.  Taking  the  proportions  before  used, 
make  Be  and  Bb,  each  on  BE,  respectively  5^  and  6^  fiftecnilis 
of  the  pitch,  to  obtain  the  point  and  root  circles  parallel  to  AB 
through  e  and  h,  since  the  real  height  be  of  the  teeth  is  shown 
in  its  real  size  on  the  extreme  element  EB  of  the  cone  EAB. 
The  horizontal  jorojections  of  these  circles  are  those  with  radii 
C'e'  and  G'b'. 

The  corresponding  inner  point  and  root  circles  are  found  by 
noting  (jr,  the  intersection  of  eC  and  GH,  and  that  of  bG  with 
GH.     This  last  point  is  horizontally  projected  at  n'. 

Having  thus  both  projections  of  all  the  circles  of  construc- 
tion: 

1°.  Lay  off  y*^  of  the  pitch,  that  is,  half  the  space  between 
two  teeth,  on  each  side  of  A',  B'  and  S,  and  from  the  points  so 
found  lay  off  the  pitch,  over  and  over,  which  will  give  all  those 
points  of  the  teeth  which  arc  m  tlip  outer  pitch-circle  A'SB'; 
and  project  these  points  on  AB. 

2^.  Througl)  the  points  just  found  on  A'SB'  draw  lines  to  C, 
limited  by  the  circle  C'w';  and  through  those  on  AB,  lines  to  E 
limited  by  ah,  for  the  outer  ends  of  the  flanks. 

y.  From  tlie  points  on  ah  draw  lines  to  C,  limited  by  the 
vertical  ])rojoctiun  of  circle  C'n',  for  the  root  lines  of  the  teeth, 
and  from  \\w.  ])()ints  llius  found,  the  inner  ends  of  the  Hanks 
radiating  from  G  and  limited  by  HL 

4°.  Make  arcs,  tangent  to  each  other  as  at  0,  Fig.  5,  Avith 
radii   EA  and   FA,   which  will  be   (l)iv.   L,   Prob.  28)  arcs  of 


PL:xvii. 


GEARING.  135 

the  developments  of  the  outer  pitch-circles  of  the  two  wheels. 
On  these  lay  ofT  the  pitch,  and  proportion  the  teeth  as  already 
described,  the  flanks  running  to  E  (below  the  border)  and  F, 
and  the  faces  drawn  with  convenient  circular  arcs  to  ]-e})lace  the 
epicycloidal  curves  OP  and  OQ.  Having  thus  found  the  width 
of  the  teeth  at  their  outer  points,  lay  off  half  this  width  on 
each  side  of  the  middle  point  of  each  tooth  on  the  circle  of 
radius  C'e'. 

5°.  Through  these  points  on  circle  C'e'  draw  lines  to  C,  lim- 
ited by  circle  Cg'  ;  project  the  points  of  circle  C'e'  upon  ec, 
vertical  projection  of  circle  Ce',  and  thence  draw  the  point  edges 
of  the  teeth  towards  C. 

6°.  Finally,  the  face  curves  at  both  ends  of  the  teeth  are 
sketched  by  hand,  tangent  to  the  flanks. 

By  precisely  similar  operations,  the  two  projections  of  the 
wheel  AD — A"D'  may  be  drawn. 

The  hub  and  arms  of  both  can  be  easily  drawn,  as  shown. 

To  become  perfectly  familiar  with  the  operations  here  de- 
scribed, work  out  the  following  variations  : 

Exercises. — 1.  Changing  the  numbers  of  teeth,  let  the  axis  of  the  wheel 
AD  be  perpendicular  to  the  paper  at  C,  so  as  to  appear  as  Fig.  4  now 
does. 

2,  Again  changing  the  number  of  teeth,  let  the  wheel  AD  be  in  gear 
with  AB  at  BH,  and  then  draw  the  figure  as  if  PI.  XVIII.  were  upside 
down,  making  C'A'SB'  the  elevation,  instead  of,  as  now,  the  plan. 

Screivs  and  terpentines. 

28.  Tri angular -tlireaded  screws. — If  the  isosceles  triangle 
cdl2,  PL  XIX.,  Fig.  1,  whose  base  is  in  the  vertical  line  Wi,  be 
revolved,  together  with  that  line,  uniforml}',  around  the  vertical 
AB  as  an  axis,  having  also  a  uniform  vertical  motion  on  E7i,  it 
will  generate  the  spiral  solid  called  the  thread  of  a  triangular- 
threaded,  also  called  a  V-threaded  screw.  The  surfaces  gen- 
erated by  (Z12  and  cl2  are  lielicoids,  upper  and  lower.  The  lines 
generated  by  the  points  c,  d  and  12  are  helices,  inner  and  outer. 
E/i  will  generate  a  cylinder,  called  the  core  or  neivel  of  the 
screw. 

29.  Square-threaded  and  other  screws. — If,  PI.  XIX.,  Fig.  2, 


136  GEARIXG, 

a  square,  EC6,  be  substituted  for  the  triangle,  the  result  will  be 
a  square-threaded  screw. 

If,  Fig.  3,  a  sphere  whose  centre  describes  a  helix  be  the  gen- 
eratrix, the  resulting  solid  will  be  that  called  a  serpentine.  This 
is  the  form  of  a  spiral  spring  formed  of  circular  wire  ;  also  of  the 
hand-rail  of  circular  stairs,  when  the  rail  has  a  circular  section 
made  by  cutting  it  "  square  across." 

Again  :  if,  Fig.  5,  the  profile  of  a  tooth  be  taken  as  the  gen- 
eratrix of  the  thread,  there  will  be  formed  the  kind  of  toothed 
wheel  called  an  endless  screw,  since  its  constant  rotation  in  one 
direction  will  actuate  tlie  wheel  L.  It  is  always  the  screw  that 
is  the  '^driver'''  and  actuates  the  wheel,  which  is  the  ^^ follower,''^ 
and  receives  a  very  slow  motion  ;  since  the  tooth  G  will  be  car- 
ried to  the  position  of  the  next  tooth  above  it,  by  one  complete 
revolution  of  the  screw. 

30.  JSfiwiber  of  threads. — In  PI.  XIX.,  Fig.  1,  one  helical  revo- 
lution of  the  generating  triangle  brings  the  side  6-12  to  the  posi- 
tion di^,  which  allows  no  intermediate  position  of  the  triangle. 
The  screw  is  therefore  single-threaded.  The  like  is  true  of  the 
screws  in  Figs.  2  and  5. 

If,  however,  Fig.  1,  one  such  revolution  of  cdl2,  had,  by 
means  of  a  greater  ascending  motion,  bror.ght  cl2  to  the  position 
rs,  the  screw  would  have  been  tivo-tltreaded;  and  if  to  the  posi- 
tion no,  it  would  have  been  three-threaded.  The  like  again  is 
true  of  other  screws,  the  number  of  threads  being  adapted  to 
the  advance  parallel  to  AB,  of  any  point  of  the  screw  in  one 
revolution.     This  advance  is  called  the  pitch  of  the  screw. 

Evidently  the  coils  of  a  second  spiral  like  Fig.  3  could  be  laid 
between  those  shown.     It  would  then  be  two-threaded. 

Ex.  0.  To  Construct  the  Projections  of  a  Triangu- 
lar-threaded Screw.     V\.  XIX.,  Fig.  1. 

The  construction  of  the  screw  consists  principally  in  that  of  its 
helices.  Accordingly,  let  AE  and  AC  be  the  radii  of  the  circles 
which  represent  the  circular  motions  of  the  points  c  and  12,  and 
which  are  the  horizontal  projections  of  the  inner  and  outer 
helices.     And  let  0,12  be  the  pitch  of  the  screw. 

As  both  component  motions,  circular  and  rectilinear,  of  the 


F  L  xvm 


GEARING.  137 

compound  helical  motion  are  uniform,  divide  the  circles  AE  and 
AC  and  the  pitch  0,12  all  into  the  same  number  of  equal  j^arts, 
here  12,  and  draw  horizontal  lines  through  the  points  of  division 
on  0,12.  Then  for  an  outer  helix,  project  C  at  0;  1  on  the  first 
horizontal  above  it ;  2  on  the  second  horizontal,  C,  at  6  on  the 
sixth  horizontal,  and  so  on  till  C  is  projected  again  at  12. 

Proceed  in  a  precisely  similar  manner,  beginning  by  jiroject- 
ing  E  at  c,  to  find  points  of  an  inner  helix.  The  lines  as  cl2 
and  dVl  complete  the  figure. 

Each  half,  to  the  right  and  left  of  AB,  of  the  visible  front 
half,  as  06,  of  an  outer  helix  is  like  the  other  half  reversed,  both 
right  for  left  and  upside  down.  Hence,  as  all  the  outer  helices  are 
alike,  the  portion  of  an  irregular  curve  which  will  fit  one  half  of 
one,  will  serve  in  ruling  them  all.  Similar  remarks  apply  to  the 
inner  helices. 

Had  the  ascent  been  from  D  to  the  left  on  the  front  half  of 
the  screw  instead  of  from  O  to  the  right,  the  screw  would  have 
been  left-handed.  Left-handed  screws  are  only  employed  for 
special  purposes,  as  when  two  rods,  placed  end  to  end,  are  to 
be  separated  or  brought  together  by  a  screw  link  working  on 
both,  as  seen  in  the  truss-rods  under  rail-car  bodies.  In  this 
case  the  screw-threads  on  one  rod  would  be  right-handed,  and 
those  on  the  other  left-handed. 

Exercises. — 1.  Construct  the  projections  of  a  two-threaded  and  of  a 
three-threaded  trianguhir  screw. 

2.  Construct  the  projections  of  a  two-threaded  and  of  a  three-threaded 
Mt-handed  screw. 

Ex.  10.  To  DraviT  a  Square-threaded  Scre-w.  PI. 
XIX.,  Fig.  2. 

The  operations  in  this  case  are  so  similar  to  those  of  the 
last  problem,  as  is  evident  from  the  figure,  that  they  need  no  de- 
tailed description.  The  form  of  the  thread  renders  the  under 
outer  helices  of  the  left  side,  and  the  iipper  outer  helices  of 
the  right  side,  of  the  screw  visible  on  the  back  half  of  the  screw 
until  they  disappear  behind  the  cylindrical  core.  Also,  the  inner 
helices  are  visible  only  on  the  under  left-hand  side  and  upper 
right-hand  side  of  the  thread. 

In  the  execution,  it  is  very  important  to  remember  that  an}/ 


138  GEAEI]S"G. 

one  helix  is,  on  the  screw  itself,  of  uniform  curvature  through- 
out, hence  though  very  sharply  curved  in  projection  at  the 
extreme  points,  as  6  and  12,  especially  in  a  single-threaded 
screw,  they  are  not  there  pointed,  except  in  drawings  on  a  small 
scale  where  they  may  be  approximately  represented  by  straight 
lines,  as  in  Figs.  7,  9,  and  10. 

Exercise. — Draw  a  square-threaded  screw  with  three  threads,  and 
show  all  four  helices  of  one  thread  throughout,  but  dotted  where 
invisible. 

Ex.  11.  To  Draw  the  Interior  of  a  Nut  or  Internal 
Screvr. 

PI.  XIX.,  Fig.  8,  shows  the  interior  of  one  half  of  the  nut 
for  a  square-threaded  screw  ;  that  is,  of  the  hollow  cylmder  with 
a  thread  on  its  interior  surface,  adapted  to  work  in  the  spaces 
between  the  threads  of  the  screw.  The  figure  representing  the 
rear  half  of  the  nut,  the  threads  must  there  ascend  to  the  left, 
as  they  do  on  the  rear  half  of  the  screw. 

Exercises. — 1.  Draw  the  vertical  section  of  the  nut  corresponding  with 
Fig.  1. 

2.  Draw  that  of  the  nut  of  a  square-threaded  screw  of  two  threads. 

Ex.  12.  To  Dravr  the  Endless  Screw  and  Worm 
Wheel.     PI.  XIX.,  Figs.  4,  5. 

The  profile  of  a  tooth  here  becomes  the  generatrix  of  a  screw- 
thread  bounded  by  helices  found  as  before.  The  pitch-line  MIST 
is  divided  by  the  pitch  as  in  the  casQ  of  a  rack,  the  pitch  of  the 
screw  and  wheel  being  the  same. 

The  wheel,  having  its  axis  in  a  direction  perpendicular  to  tluit 
of  the  screw,  is  in  reality  a  short  piece  of  a  screw  having  a  very 
great  pitch.  That  is,  the  angle  made  by  the  helices  of  the 
wheel-teeth  with  a  i)lane  perpendicular  to  the  axis  of  the  wheel, 
that  is,  with  the  plane  of  the  paper,  is  the  complement  of  the 
angle  made  by  the  screw  helices  with  a  plane  perpendicular  to 
its  axis,  that  is,  to  a  plane  perpendicular  to  the  paper  on  GD. 
The  curves,  as  that  to  the  left  of  N,  which  represent  the  further 
ends  of  the  teeth,  are  assumed,  unless  the  width  of  the  wheel  is 
shown  by  a  plan  view.. 


GEARING.  139 

Ex.  13.   To  Draw  a  Serpentine.    PI.  XIX.,  Fig.  3. 

This  surface  is  one  which,  like  a  thin  helical  tube,  would  in- 
close, tangentially,  all  the  positions  of  a  sphere,  indicated  by 
the  dotted  circles,  whose  centre  should  describe  a  helix,  ACB — 
2345. 

The  contours,  or  apparent  bounding  lines,  of  the  serpentine  are 
not  helices,  though  at  a  uniform  perpendicular  distance  from  the 
central  helix,  but  are  drawn  tangent  to  the  numerous  equal 
dotted  circles  having  their  centres  on  the  helix,  and  which  repre- 
sent as  many  positions  of  the  generating  sphere. 

Surfaces  which,  like  the  sphere  and  serpentine,  are  nowhere 
straight,  are  call  double-curved.  Where  partly  convex,  as  on  the 
outer  circle,  or  in  the  circle  OF,  and  partly  concave,  as  on  the 
inner  side,  or  on  the  circle  OD,  the  contour  vanishes  into  the 
surfaces,  at  certain  points,  when  shown  by  a  line  drawing,  as  is 
seen  at  the  left  of  the  under  contours,  and  the  right  of  the 
upper  ones. 

The  lower  coil  is  shown  approximately  as  straight,  indicating 
jfhat  would  be  permissible  in  rough  drawings  or  on  a  small  scale. 


DIVISION     SIXTH. 

SIMPLE  STRUCTURES  AND  MACHINES. 


'23  7.  Note.  The  objects  of  this  Division"  are,  to  acquaint  the 
student  with  a  few  things  respecting  the  drawing  of  whole  structures 
which  are  not  met  with  in  the  drawing  of  mere  details ;  to  serve  as 
a  sort  of  review  of  practice  in  certain  processes  of  execution ;  and 
to  afford  illustrations  of  }»arts  of  structures  whose  names  have  yet 
to  be  defined.  Proceedino;  with  the  same  order  as  regards  material 
that  was  observed  in  Division  Second,  we  have  : — 

CHAPTER  I. 

STONE   STRUCTUKES. 

238.  Example  1°.  A  brick  segmental  Arch.  I'l.  XX.,  Fig.  123. 

Description  of  the  structure. — A  segmental  arch  is  one  whose 
curved  edges,  as  aCc,  are  less  than  semicircles.  A  brick  segmental 
arch  is  usually  built  Avith  tlie  widths  of  the  bricks  placed  radially, 
since,  as  the  bricks  are  rectangular,  the  mortar  is  disposed  between 
tliem  in  a  wedge  form  in  order  that  each  brick  with  the  mortar 
attached  may  act  as  a  wedge ;  while  if  the  length  of  the  bricks  be 
radial,  the  mortar  spaces  will  be  inconveniently  wide  at  their  outer 
ends,  unless  the  arch  be  a  very  wide  one,  or  unless  it  have  a  very 
large  radius. 

The  2:)ermanent  sujjports  of  the  arch,  as  nPT,  are  called  abut 
ments^  and  the  radial  surface,  as  nab^  against  which  the  arch  rests, 
is  called  a  skew-hack. 

The  temporary  supports  of  an  arch  wliile  it  is  being  built  are 
called  centres  or  centrings.,  and  vary  fi-om  a  mere  curved  frame 
made  of  jiieces  of  board — as  used  in  case  of  a  small  drain  or  round 


PL.XIX. 


STONE    STKUCTUUES.  J  4 1 

topped  window — to  a  heavy  and  complicated  fianiini,',  as  used  foi 
the  temporary  support  of  heavy  stone  bridejes. 

Note.  The  general  designing  of  these  massive  centrings  may 
rail  for  as  much  of  scientific  engineei'ing /;«o?cZec?(/e,  and  their  details 
and  management  may  call  for  as  much  practical  engineering  skill, 
as  does  the  construction  of  the  permanent  works  to  which  these 
centrings  are  auxiliary.  In  short,  the  detailed  design  and  manage- 
ment of  auxiliary  constructions,  in  general,  is  no  unimportant  depart- 
ment of  engineering  study. 

The  span  is  the  distance,  as  ac,  between  the  points  of  support,  on 
the  under  surface  of  the  arch.  The  stones  over  the  arch  and  abut- 
ment, form  the  spandril^  or  hacking^  Qc^P. 

239.  G-rapli'tcal  construction. — Let  the  scale  be  one  of  four  feet 
to  the  inch=48  inches  to  one  inch=:j\.  Draw  RT  to  represent 
tlie  horizontal  surface  on  which  the  arch  rests.  Let  the  radius  of 
the  inner  curve  of  the  arch  be  V  feet,  the  height  of  the  line  ac  from 
the  ground  2  feet  8  inches,  and  the  span  7  feet.  Then  at  some 
point  of  the  ground  line,  draw  a  vertical  line,  OC,  for  a  centre 
line  ;  then  draw  the  abutments  at  equal  distances  on  each  side  of 
the  centre  line,  and  6  feet  8  inches  aj)art.  Let  them  be  2  feet  6 
inches  wide. 

Since  the  span  and  radius  have  been  made  equal,  Oh  and  Oc?  may 
be  drawn,  in  this  example,  with  the  60"  triangle.  Drawing  these 
lines,  and  making  Oa  —  7  feet,  make  ab  =  one  foot,  draw  the  two 
curves  at  the  end  of  the  arch,  and  make  b  and  d  points  in  the  top 
surfaces  of  the  abutments. 

To  locate  the  bricks,  since  the  thickness  of  the  mortar  between 
the  bricks,  at  the  inner  curve  of  the  arch,  would  be  very  slight,  lay 
off  two  inches  on  the  arc  aCc  an  exact  number  of  times.  The  dis- 
tance taken  in  the  compasses  as  two  inches,  may  be  so  adapted  aa 
to  be  contained  an  exact  number  of  times  in  aCc,  since  the  thick- 
ness of  the  mortar  has  been  neglected,  but  would  in  practice  be  so 
adjusted,  as  to  allow  an  exact  number  of  whole  bricks  in  each 
course. 

The  arch  being  a  foot  thick,  there  will  be  three  rows  of  bricka 
seen  in  its  front.  Draw  therefore  two  arcs,  dividing  ab  and  cd  into 
three  spaces  of  four  inches  each,  and  repeat  the  process  of  division 
on  both  of  them. 

Having  all  the  above-named  divisions  complete,  fasten  a  fina 
needle  vertically  at  O,  and,  keeping  the  edge  of  the  ruler  against  it, 
to  keep  that  edge  on  the  centre  without  difiiculty,  draw  the  lines 
which  represent  the  joints  in  each  of  the  three  courses  of  brick. 


142  STON^E   STRUCTURES. 

240.  Ex.  2®.  A  semi-cylindrical  Culvert,  having  vertical 
quarter-cylindrical  Wing  Walls,  truncated  obliquely.  PI 
XX.,  Fig.  124. 

Description  of  the  structure, — A  culvert  is  an  arched  passage, 
often  tlat  bottomed,  constructed  for  the  purpose  of  carrying  water 
under  a  canal  or  other  thoroughfare.  Wing  walls  are  curved  con- 
tinuations of  the  vertical  fiat  Avail  in  which  the  end  of  the  arch  is 
seen.  Their  use  is  to  support  tlie  embankment  through  which  the 
culvert  is  made  to  pass,  and  to  prevent  loose  materials  I'rom  the 
embankment  fx'om  working  their  way  or  being  washed  into  the  cul- 
vert. Partly,  perhaps,  for  appearance's  sake,  the  slope  of  the  plane 
which  truncates  the  flat  arch-wall,  called  the  spandrll  wall,  and  the 
wing  walls,  is  parallel  to  the  slope  of  tlie  embankment.  The  wing 
walls  are  often  terminated  by  rectangular  flat-topped  posts — "  piers" 
or  "buttresses,"  AA',  and  the  tops,  both  of  these  piers  and  of  the 
walls,  are  covered  with  thin  stones,  abed — a"b"c"d\  broader  than 
the  wall  is  thick,  and  collectively  called  the  coping. 

Since  the  parts  of  stone  structures  are  not  usually  firmly  bound 
or  framed  together,  each  course  cannot  be  regarded  as  one  solid 
piece,  but  rather  each  stone,  in  case,  for  instance,  of  the  lowermost 
course,  rests  directly  on  the  ground, independently  of  other  stones 
of  the  same  course,  hence  if  the  ground  were  softer  in  some  spots, 
under  such  a  course,  than  in  others,  the  stone  resting  on  that  spot 
would  settle  more  than  others,  causing,  in  time,  a  general  disloca- 
tion of  the  structure.  Hence  it  is  important  to  have  what  are  called 
continuous  bearings,  that  is,  virtually,  a  single  solid  piece  of  some 
material  on  which  several  stones  may  rest,  and  placed  between 
the  lowest  course  and  the  ground. 

Timbers  buried  away  from  the  air  are  nearly  imperishable  ;  hence, 
tiiiiljcrs  laid  upon  the  ground,  if  that  be  fii-m,  and  covered  with  a 
double  floor  of  plank,  form  a  good  foundation  for  stone  structures; 
and  in  the  case  of  a  culvert,  if  such  a  flooring  is  made  continuous 
over  the  whole  space  covered  by  the  arch,  it  will  prevent  the  flow- 
ing water  from  washing  out  the  earth  under  the  sides  of  the  arch. 

When  the  wing  walls  and  spandril  are  built  in  courses  of  uni- 
form thickness,  the  arrangement  of  the  stones  forming  the  arch, 
so  as  to  bond  neatly  with  those  of  the  walls,  offers  some  difficul- 
ties, as  several  things  are  to  be  harmonized.  Thus,  the  arch  stones 
must  be  of  equal  thickness,  at  least  all  except  tlie  top  one,  and  then, 
there  must  be  but  little  difference  between  the  widths  of  the  top, 
or  key  stone,  and  the  other  stones  ;  the  stones  must  not  be  di.s- 
projiortionately  thin  or  very  wide,  they  .should  have  no  re-entranf 


STONE    STnUCTURES.  143 

angles,  or  very  acute  angles,  and  there  must  not  be  any  groal 
extent  of  unbroken  joint. 

241.  Grapliical  constmiction. — Let  the  scale  be  that  of  five  feet 
to  an  inch  =  60  inches  to  an  inchrz:^'^, 

a.  Draw  a  centre  line,  15B',  for  the  plan, 

h.  8nj)posing  the  radius  of  the  outer  surface,  or  back,  of  the  arch 
to  be  b\  feet,  draw  CC  parallel  to  BB'  and  5^  feet  from  it. 

c.  Draw  BE,  and  on  CC  produced,  make  EC  =  9  feet  8  inches, 
CD,  the  thickness  of  the  face  wall  of  the  arch  =  2  feet  4  inches,  and 
the  radius,  oD,  of  the  lace  of  the  wing  wall=4  feet. 

d.  With  o  as  a  centre,  draw  the  quadrants  CG,  and  DF,  and  Avith 
a  radius  of  3  feet  8  inches,  draw  the  arc  oA,  the  plan  of  the  inner 
edge  of  the  coping.  Also  draw  at  D  and  C,  lines  perpendicular 
to  BB'  to  represent  the  face  Avail  of  the  arch. 

e.  At  G,  draw  G/i  towards  o,  and  =3  feet,  for  the  length  of 
the  cap  stone  of  the  buttress,  A  A',  and  make  its  Avidth  =2  feet 
10  inches,  tangent  to  CG  at  G.  The  top  of  this  cap  stone,  being  a 
flat  quadrangular  pyramid,  draw  diagonals  through  G  and  A,  to 
represent  its  slanting  edges. 

f.  Supposing  the  arch  to  be  \\  feet  thick,  make  C'II=1^  feet, 
and  at  C  and  H,  draw  the  irregular  curved  lines  of  the  broken  end 
of  the  arch,  and  the  broken  line  near  the  centre  line,  also  a  fragment 
of  the  straight  part  of  the  coping. 

g.  Let  the  horizontal  course  on  Avhich  the  arch  rests,  be  2  feet  9 
inches  wide,  i.e.,  make  He  =  3  inches,  and  CW=1  foot;  and  let  the 
planking  project  3  inches  beyond  the  said  course,  making  e;-=3  feet. 
Through  e,  n  and  r,  draAV  lines  parallel  to  BB'  and  extending  a  little 
to  the  right  of  C'll. 

h.  Proceeding  to  represent  the  parts  of  the  arch  substantially  in 
the  order  of  their  distance  from  the  eye,  as  seen  in  a  plan  A^iew,  a 
portion  of  the  planking  may  next  be  represented.  The  paira  of 
broken  edges,  and  the  position  of  the  j(jints,  shoAv  that  there  are 
two  layers  of  plank  and  that  they  break  joints. 

i.  Under  these  planks,  appear  the  foundation  timbers,  Avliich 
being  laid  transversely,  and  being  one  foot  wide  and  one  foot  apart, 
are  represented  by  parallels  one  foot  apart,  and  perpendicular  to 
BB'.  Let  the  planking  project  4  inches  beyond  the  left  hand  tim 
ber.  Observe  that  tAVO  timbers  touch  each  other  under  the  arch 
front. 

j.  The  general  arrangement  of  stones  in  the  curved  courses  of  the 
^ing  wall,  in  order  that  they  may  break  joints,  is,  to  have  three  and 
four  stones,  respectively,  in  the  consecutive  courses.     To  indicate 


J  44  STONE    STliUCTUKES. 

this  arrangement  in  tbe  plan,  hG,f//,  (^(^and  DC  will  represent  tJQf 
joints  of  alternate  courses,  and  the  lines  km,  &g.  midway  betweec 
the  former,  will  represent  the  joints  of  the  remaining  interraediato 
toursos. 

Tliis  completes  a  partial  and  dissected  plan  which  shows  more  of 
the  construction  than  would  a  plan  view  of  the  finished  culvert,  and 
a=  much,  as  if  the  j^arts  on  both  sides  of  the  centre  line  were  shown. 
In  fact,  in  drawings  which,  are  strictly  working  drawings,  each  i)ro- 
jection  should  show  as  much  as  possible  in  regard  to  each  distinct 
part  of  the  object  represented. 

242.  Passi7ig  to  tJie  side  elevation,  which  is  a  sectional  one,  show- 
ing parts  in  and  beyond  a  vertical  plane  through  the  axis  of  the 
arch,  we  have  : — 

a.  The  foundation  timbers,  as  rn'q,  &c.,  projected  up  from  the 
plan ;  or,  one  of  them  being  so  projected,  the  others  may  be  ecu- 
Btructed,  independently  of  the  plan,  by  the  given  measurements. 

b.  The  double  course  of  planking  ojo,  appears  next  with  an  occa- 
sional vertical  joint,  showing  where  a  plank  ends. 

'•.  The  buttress,  A,  and  its  cap  stone  Y,  are  projected  up  from 
tiie  plan,  and  made  6  feet  high,  from  the  planking  to  G'. 

d.  From  G'  and  A',  the  slanting  top  of  the  wing  walls  are  shown, 
as  having  a  slope  of  1^  to  1 — i.  e.  //,7t"=f  lo'u — and  the  vertical  lines 
at  C,  D'  and  D"  are  projected  up  from  C,  D  and  D'". 

The  remaining  lines  of  the  side  elevation  are  best  projected  back 
from  the  end  elevation,  when  that  shall  have  been  drawn. 

243.  In  the  end  or  front  elevation,  we  have: — 

a.  At  m''m"',  a  side  view  of  one  of  the  foundation  timbers, 
broken  at  vi'",  so  as  to  show  other  timbers  behind  it. 

b.  The  planking  o'o"  in  this  view,  shows  the  ends  of  the  planks 
in  both  layers — breaking  joints. 

c.  Nt>'  =  Bo"',  taken  from  the  plan  ;  and  in  general,  all  the  hori- 
zontal distances  on  this  elevation,  are  taken  from  the  plan,  on  linea 
perpendicular  to  BB'. 

d.  Tlie  vertical  sides  of  the  buttress,  A',  are  thus  found.  Tho 
heights  of  its  parts  are  projected  over  from  the  side  elevation. 

e.  The  thickness  of  the  foundation  course,  ts=\\  feet,  and  tr'—en, 
on  the  plan. 

/.  The  centre,  O,  of  the  face  of  the  arch,  is  in  the  line  r't  pro- 
duced. The  radius  of  the  inner  curve  (intrados)  of  the  arch  is  4  feet 
»nd  of  the  cylindrical  back,  behind  the  lace  wall,  5}  feet — shown  by 
a  dotted  arc.  In  representing  the  stones  forming  the  arch,  it  is  to 
be  remembered  that  they  must  be  equal,  except  the  "  key  stone," 


STONE   STKUCTURE^>.  I  <5 

^,  which  may  be  a  little  thicker  than  the  others;  they  must  also  be 
of  agreeable  proportions,  free  from  very  acute  angles,  or  from  re- 
entrant obtuse  angles ;  and  must  interfere  as  little  as  possible  with 
the  bond  of  the  regular  horizontal  courses  of  the  whig  walls.  There 
must  also  be  an  odd  number  of  stones  (ring  stones)  in  the  front  of 
the  ai'ch. 

On  both  elevations,  draw  the  horizontal  lines  representing  th 
«ing  wall  courses  as  one  foot  in  thickness,  and  divide  the  inner 
curve  of  the  arch  into  15  equal  parts.     Draw  radial  lines  tlirough 
the  points  of  division.     Their  intersections  with  the  horizontal  lines 
are  managed  according  to  the  principles  just  laid  down. 

g.  The  points,  as  h  and/",  in  the  plan,  arc  then  projected  into  the 
alternate  courses  of  the  side  elevation,  and  into  the  line,  Bo'",  of 
the  plan. 

From  the  latter  line,  the  several  distances,  o"'h"\  &c.,  from  o"\ 
thus  found,  are  transferred  to  the  line  o'N,  as  at  o'b"'\  (fee,  and  at 
these  points  the  vertical  joints  of  the  front  elevation  are  drawn  in 
their  proper  position,  as  being  the  same  actual  joints,  shown  by  the 
veitical  lines  of  the  side  elevation.  In  the  stones  immediately  under 
the  coping,  there  must  generally  be  some  irregularity,  in  order  to 
avoid  triangular  stones,  or  stones  of  inai>propriate  size. 

//.  To  construct  the  front  elevation  of  the  coping.  All  points,  aa 
a,  a',  a",  in  either  the  front  or  back,  or  upper  or  lower  edges  of  the 
co])ing,  are  found  in  the  same  way,  and  as  follows : 

a"  is  in  a  horizontal  line  through  a'  and  in  a  line  a"a"'\  whose 
distance  from  o'  equals  the  distance  o"'a"'  on  the  plan.  Construct- 
ing other  points  similarly,  the  edges  of  the  coping  may  be  drawn 
with  an  "  irregular  curve." 

The  horizontal  portion  of  the  coping,  over  the  arch,  is  projected 
over  from  C  and  from  the  two  ends  of  the  vertical  line  at  D'. 

Execution. — In  respect  to  this,  the  drawing  explains  itselC 

Example.  Let  this  design,  or  any  similar  one,  be  drawn  on  a 
scale  of  four  fest  to  the  inch,  on  a  larger  plate;  not  forgetthig  to 
place  tlie  three  projections  in  their  proper  relative  position,  as 
shown  (15)  and  (32). 


1 


CHAFrER   II. 

WOODEN    sxnucTunES. 

244.  Ex.  3».    Elevation  of  a  "  King  Post  Truss." 
Mechanical  constmction,  <£c. — A  Truss  is  an  assemblage  of  pieces 

90  fastened  together  as  to  be  virtually  a  single  piece,  and  therefore 
exerting  only  a  vertical  force,  due  to  its  weight,  upon  the  support- 
ing walls. 

In  PL  XX.,  Fig.  125,  A  is  a  tie  beam/  B  is  a. pri7icipal /  C  is  a 
rafter /  D  is  the  kin// post/  E  is  a  strut  /  F  is  a  tcall plate  /  G  is  a 
purlin — running  parallel  to  the  ridge  of  the  roof,  from  truss  to  truss, 
and  supporting  the  rafters.  H  is  the  ridgepole;  AV  is  the  wall^ 
and  ab  is  a  strap  by  which  the  tic  beam  is  suspended  from  the  king 
post. 

245.  Grraphical  constrnction. — In  the  figure,  only  half  of  the  truss 
is  shown,  but  the  directions  apply  to  the  drawing  of  the  whole.  In 
these  directions  an  accent,  thus  '  ,  indicates  feet,  and  two  accents, 
' ,  inches.  For  ])ractice  draw  the  whole  Ugure,  and  on  a  larger  scale. 

a.  Draw  the  vertical  centre  line  6D. 

b.  Draw  the  upper  and  lower  edges  of  the  tie  beam,  one  foot 
apart,  and  12'  in  length,  on  each  side  of  the  vertical  line. 

c.  On  the  centre  line,  lay  off  from  the  top  of  the  tie  beam,  5' — G' 
to  locate  the  intersection  of  the  tops  of  the  principals ;  and  on  the 
top  of  the  tie  beam,  lay  off  II'  on  each  side,  to  locate  the  intersec- 
tion of  the  upper  faces  of  the  principals  with  the  top  of  the  tie  beam. 

d.  Draw  the  line  joining  the  two  points  just  found,  and  on  any 
perpendicular  to  it,  as^]/,  lay  off  its  depth  =  8",  and  draw  its  lower 
edge  parallel  to  the  ui)per  edge.  Make  the  shoulder  at  0  =  3'  and 
parallel  tofg. 

e.  From  tlie  top  of  the  beam,  draw  short  indefinite  lines,  c,  0* 
each  side  of  the  centre  line,  and  note  the  points,  as  c,  where  they 
would  meet  the  upper  sides  of  the  principals. 

f.  Draw  vertical  lines  on  each  side  of  the  centre  line  and  4' 
from  it. 

g.  From  tlie  points,  as  e,  draw  lines  parallel  to  fg  till  they  inter- 
«ect  the  last  named  vertical  lines. 


•WOODEN   STEUCTURKS.  147 

h.  Jviake  ns  =  o' — 9".  Make  the  short  vertical  distance  at  c-.=4' 
draw  so,  and  make  tlie  upper  side  of  the  strut  parallel  to  sc,  and  4* 
from  it.  Note  the  intersection  of  this  parallel  with  the  line  to  the 
left  of  D,  and  comiect  this  point  with  the  upper  end  of  c,  to  com- 
plete the  strut. 

i.  Draw  the  edges  of  the  rafter,  parallel  to  those  of  the  principal, 
4'  apart,  and  leaving  4"  between  the  rafter  and  the  principal.  At 
Of  draw  a  vertical  line  till  it  meets  the  lower  edge  of  C,  and  from 
this  intersection  draw  a  horizontal  line  till  it  meets  the  upper  edge 
of  C ;  Avhich  gives  proper  dimensions  to  the  wall  plate. 

J.  From  the  intersections  of  the  upper  edges  of  the  rafters,  lay 
off  downwards  on  the  centre  line  12",  and  make  the  ridge  pole, 
thus  located,  3"  Avide. 

k.  In  the  middle  of  the  upper  edge  of  the  principal,  place  the 
purlin  4"  x  6",  and  setting  2"  into  the  principal. 

I.  Let  the  strap,  ab,  be  2"  wide,  and  2' — 6"  long  from  the  bottom 
of  the  tie  beam.  Let  it  be  spiked  to  the  king  post  and  tie  beam, 
and  let  it  be  half  an  inch  thick,  as  shown  below  the  beam.  W,  the 
supportmg  wall,  is  made  at  pleasure. 

Execution. — This  mainly  explains  itself.  As  working  drawmga 
usually  have  the  dimensions  figured  upon  them,  let  the  dimensions 
be  recorded  in  small  hair  line  figures,  between  arrow  heads  which 
denote  what  points  the  measurements  refer  to. 

246.  Ex.  4«.  A  "Queen  Post  Truss"  Bridge.  PI.  XXL, 
Fig.  126. 

Mechcmical  construction. — ^This  is  a  bridge  of  33  feet  span,  ovei 
a  canal  20' — 6*  wide  between  its  banks  at  top,  and  20' — 2"  at  the 
water  line.  It  rests  on  stone  abutments,  R  and  P,  )ne  of  \\hich  ia 
represented  as  resting  on  a  plank  and  timber  foundation,  the  other 
on  "  piles." 

A  is  the  tie  beam  ;  B,  B'  the  queen  posts  ;  C,  C  the  principals; 
D  the  collar  beam.,  or  straining  sill  ^  R,  P,  the  abutments  ;  eQ,t  the 
pavement  of  the  tow  path ;  iK.  the  stone  side  walls  of  the  canal; 
TP  the  opposite  timber  wall,  held  by  timbers  UU',  N,  dovetailed 
into  the  wall  timbers ;  E,  S,  the  piles,  iron  shod  at  bottom.  These 
are  the  principal  parts. 

247.  Graphical  construction. — Let  the  scale  be  one  of  five  feet 
to  the  inch. 

a.  All  parts  of  the  truss  are  laid  off  on,  or  from,  the  centre  line 
AD.  A  is  14"  deep;  the  dimensions  of  BB'  are  12' x 6',  except  at 
top,  where  they  are  10"  wide  for  a  vertical  space  of  16".     C  and  D 


148  WOODRX    STRUCTURE8. 

are  each  10'  deep.  BB'  are  10'  apart,  and  the  feet  ofC  and  C,  12* 
from  the  ends  of  the  tie  beam,  which  is  36'  long.  D  is  6"  below 
the  top  of  the  queen  posts,  rr  are  inch  rods  with  five  inch  wasliers, 
\"  thick,  and  nuts2^''xl'.  6i' is  a  f  bolt;  with  washer  4"x?' 
and  nuts,  2''xl';  and  perpendicular  to  the  joint,  ad. 

b.  From  each  end  of  the  tie  beam,  lay  oflf  1' — 9"  each  way  for  the 
width  of  the  abutments,  at  the  top.  Make  the  right  hand  abutment 
rectangular  in  section  and  11'  high,  of  rectangular  stones  in  irregu- 
lar bond  (70).  Let  the  left  hand  abutment  have  a  batter  of  1"  in  1' 
on  the  side  towards  the  canal,  and  let  it  be  eleven  feet  high,  in 
eleven  equal  courses. 

c.  Make  et^  the  width  of  the  paved  tow  path  =  7' — 6',  with  a  rise 
in  the  centre,  at  Q,  of  6". 

d.  The  side  wall  is  of  rubble,  4'  thick  at  bottom,  and  extenduig 
18"  below  the  water,  with  a  batter  of  1"  in  1',  and  having  its  upper 
edge  formed  of  a  timber  12"  square. 

e.  The  right  hand  abutment  rests  on  a  double  cour.se  of  three-inch 
planks,  qq\  5^'  broad,  and  resting  on  four  rows  of  10"  pUes,  ES. 
S  is  the  sheet  iron  conical  shoe  at  the  lower  end  of  one  of  these 
piles,  the  dots  at  the  upper  end  of  which  rejiresent  nails  which 
fasten  it  to  the  pile. 

f.  IT  is  a  timber  wall  having  a  batter  of  l"  to  1',  and  held  in 
place  by  timbers,  UU',  N,  dovetailed  into  it  at  its  horizontal  joints, 
hi  various  places. 

(J.  The  water  line  is  2'  below  T<,  and  the  water  is  4^  feet  deep. 

248.  Execution. — It  is  intended  that  this  plate  should  be  tinted, 
though,  on  account  of  the  difficulty  of  procuring  adequate  engraved 
fac-similes  of  tinted  hand-made  drawings,  it  is  here  shown  only  aa 
a  finished  line  drawing,  and  as  such,  explains  itself,  after  observing 
that  as  the  left  hand  abutment  is  shown  in  elevation,  it  is  dotted 
below  the  ground  ;  while,  as  the  right  hand  abutment  is  shown  in 
Bection,  it  is  made  wholly  in  full  lines,  and  earth  is  shown  only  at 
each  side  of  it. 

The  usual  conventional  rule  is,  to  fill  the  sectional  elevation  of  a 
fctonc  wall  with  wavy  lines;  but  where  other  niaiks  servo  to  distin- 
guish elevations  from  sections,  as  in  the  case  just  described,  this 
labor  is  unnecessary. 

The  following  would  be  the  general  order  of  operations,  in  case 
this  drawing  weie  shaded. 

a.  Pencil  all  parts  in  fine  faint  lines. 

h.  Ink  all  paits  in  fine  lines. 

c.  Grain  the  wood  work  with  a  very  tine  pen  and  light  indi/xn  ink, 


^^tJ^^^^^^J^gi^ 


PL.XX-. 


WOODEX    STRUCTURES. 


149 


the  sides  of  timbers  as  seen  on  a  ncAvly-planed  board,  the  ends  oi 
large  timbers  in  rings  and  radial  cracks,  and  the  ends  of  planks 
in  di3.gonal  straight  lines.  See  also  the  figures  at  y,  where  the 
lines  of  graining  outside  of  the  knots,  are  to  extend  throughout 
the  tie  beam. 

d.  Tint  the  wood  work — the  sides  with  pale  clear  burnt  sienna, 
the  ends  with  a  darker  tint  of  burnt  sienna  and  indian  ink. 

e.  Tint  the  abutments,  and 
other  stone  work,  with  prussian 
blue  mixed  with  a  little  carmine 
and  indian  ink,  put  on  in  a  very 
light  tint. 

/.  Grain  the  abutments  in 
waving  rows  of  fine,  pale,  verti- 
tical  lines  of  uniform  thickness, 
about  one  sixteenth  of  an  inch 
long,  leaving  the  uj)per  and  left- 
hand  edges  of  the  stones  blank, 
to  represent  the  mortar.  The 
part  of  the  left-hand  abutment 
which  is  under  ground  is  dotted 
only,  as  in  the  plate. 

g.  Grain  the  canal  walls  and 
paving,  as  shoAvn  in  the  plate, 
to  indicate  boulder  rubble. 

h.  Shade  the  piles  roughly, 
they  being  roughly  cylindrical; 
tint  them  with  pale  burnt  sien- 
na, and  the  shoe,  S,  with  prus- 
sian blue,  the  conventional  tint 
for  iron. 

i.  Eule  the  water  in  blue  lines, 
distributed  as  in  the  figure. 

j.  Tint  the  dirt  in  fine  horizon- 
tal strokes  of  any  dingy  mixture. 

Note. — The  above  figure  shows  a  little  more  than  half  of  a  queen-post 
ro^-truss  of  43  feet  span.  Omitting  the  light  upper  pieces,  it  may  serve 
in  place  of  Fig.  126  as  a  longer  bridge  truss;  and  may  be  drawn  on  any 
convenient  scale  ivova.four  to  six  feet  to  an  inch. 


150  WOODEX    STKUCTUEES. 

in  which  burnt  sienna  prevails,  in  the  parts  above  the  water, 
and  ink,  in  the  muddy  parts  below  the  water,  and  then  add,  oi 
not,  the  pen  strokes  shown  in  the  plate,  to  represent  sand, 
gravel,  &c. 

k.  Place  heavy  lines  on  the  right-hand  and  lower  edges  of  all 
surfaces,  except  where  such  lines  form  dividing  lines  between 
two  surfaces  in  the  same  plane.  A  heavy  line  on  the  under  side 
of  the  floor  planks,  indicates  that  those  planks  project  beyond 
the  tie  beam  A. 


~K. 


■D 


PL.XXI, 


My<^'^y'sv.Tiy^-y„j,'.   t/i.m.t.  AiCTtig 


art'   BIllUOlll   oxivi  oiuaix   iiiuv  vuv/   j7ic»v».t 


CHAPTER  m. 

IRON    CONSTEUCnONS. 

262.  Kx.  6".    A  Railway  Track.     PI.  XXII.,  Figs.  129-13-i. 

Mechanical  construction^  tbc. — It  may  be  thought  an  oversight 
to  style  tliis  plate  the  drawing  of  a  railroad  track ;  but  taking  the 
track  alone,  or  separate  from  its  various  special  supports,  as  bridges, 
&c.,  its  graphical  lopresentation  is  mainly  summed  up  in  that  of  two 
parts  ;  first,  the  union  of  two  rails  at  their  joints;  second,  the  inter- 
section of  two  rails  at  the  crossmg  of  tracks,  or  at  turn-outs.  The 
fixture  shown  in  Fig.  129,  placed  at  the  intersection  of  two  rails  tc 
allow  the  unobstructed  passage  of  car  wheels,  in  either  direction  on 
either  rail,  is  called  a  "  Frog."  Let  y  and  z  be  fragments  of  two 
rails  of  the  same  track,  then  the  side  H/'  of  the  point  of  the  frog, 
and  the  portion  k  k'  of  its  side  flange,  B,  are  in  a  line  with  the 
edges,  denoted  by  dots,  of  the  rails  y  and  s,  so  that  as  the  wheel 
passes  either  way,  its  flange  rolls  through  the  groove,  I,  without 
obstruction.  When  the  wheel  passes  from  y  towards  z  there  is  a 
possibility  of  the  flange's  being  caught  in  the  groove,  J,  by  dodg- 
ing the  point,/*.  To  guard  against  this,  a  guard  rail,  g g,  is  placed 
near  to  the  inside  of  the  other  rail,  supposed  to  be  on  the  side  of 
the  frog  towards  Fig.  132,  as  shown  in  the  small  sketch,  Fig.  132, 
which  prevents  the  pair  of  wheels,  or  the  car-truck,  from  working 
BO  far  towards  the  flange,  B,  as  to  allow  the  flange  of  the  wheel  to 
run  into  the  groove,  J,  and  so  run  off"  the  track.  F/",  and  the  por- 
tion, I  /',  of  the  flange.  A,  arc  in  a  line  with  the  inner  edge  of  the 
rail  of  a  turn-out,  for  instance,  the  opposite  rail  being  on  the  side  of 
the  frog  towards  the  upper  border  of  the  plate,  as  shown  in  Fig. 
132.  Hence  the  flange  of  a  car  wheel  in  passing  in  either  direc- 
tion on  the  turn-out,  passes  through  the  groove,  J,  and  is  prevented 
from  running  into  the  groove,  I,  by  a  guard  rail,  near  the  inner 
edge  of  the  opposite  turn-out  rail,  as  at  U,  Fig.  132. 

253.  Fig.  130  represents  the  under  side  of  the  right  hand  portion 
of  the  frog,  and  shu^v8  the  nuts  which  secure  one  of  the  bolts  which 
aecuve  the  steel  plates,  as  D,  E ;  bolts  whose  heads,  as  at  u  and  v, 
are  smooth  and  sunk  into  the  plates  so  that  their  upper  surfaces  ar« 


153  ''^ON    CONSTRUCTIONS. 

flush.  It  will  be  seen  that  there  aie  two  imts  on  each  bolt,  aa  at 
D',  on  the  bolt  u — DD',  which  appears  below  the  elevation,  since  it 
occurs  between  two  of  the  cross-ties  (sleepers)  of  the  track.  The 
nuts,  as  L,  belonging  to  the  bolt,  b",  which  are  in  the  chaiis,  q'p\ 
to',  x',  are  sunk  in  cylindrical  recesses  in  the  bottom  of  the  frog,  so 
as  not  to  interfere  with  the  cross-tie  on  which  the  surliice,  L,  rests. 
The  extra  init  is  called  a  check  or  "jam"  nut.  When  screwed  on 
snugly  it  wedges  the  first  nut  and  itself  also  against  the  threads  of 
the  screw,  so  that  the  violent  tremulous  motion  to  which  the  frog 
is  subjected  during  the  rapid  passage  of  heavy  trains  cannot  start 
either  of  them. 

Li  the  end  elevation.  Fig.  131,  A  is  the  recess  in  the  chair  x  x\ 
fitted  for  the  reception  of  the  rail,  and  B  is  the  end  of  a  rail  in  its 
place,  as  shown  at  y  in  the  plan. 

254.  Graphical  Construction. — From  the  above  description  it 
follows  that  the  whole  length  of  the  frog  depends  on  the  shape  of 
the  part  11/*  F,  and  the  distance  between  this  part  and  the  side 
rails,  as  c  ?.  In  the  piesent  example  a  c  =  l' — 11"  and  cf  =2  20".  ed 
is  11"  and  nk  is  2' from  Ff.  Having  these  relations  given,  and 
knowing  that  the  lines  at  the  extreme  ends  are  perpendicular  to  the 
rails  at  those  ends,  the  several  figures  of  the  frog  can  be  constructed 
from  the  given  measurements,  without  further  explanation. 

255.  The  construction  of  railway-track  joints  so  as  to  secure  as 
nearly  as  possible  the  uniform  firmness  of  a  continuous  rail,  has 
long  exercised  the  minds  of  railway  inventors.  Cast-iron  chairs, 
wrouglit-iron  chairs,  long  chairs  resting  on  ties  each  side  of  the 
joint,  compound  rails  (Div.  11. ,  140)  solid -headed,  or  split 
through  their  entire  height,  and  fish-joints  have  all  been  used ; 
several  of  them  in  various  forms.  Fig.  133  is  an  isometrical 
drawing — scale  -^ — of  a  wooden  fish-joint  which  allows  great 
smoothness  of  "toiotion  and  freedom  from  the  loud  clack  which 
accompanies  the  use  of  ordinary  chairs.  A,  A,  A,  are  the  sleep- 
ers (cross-tics),  D  is  a  stout  oak  plank,  perhaps  six  feet  long, 
resting  on  three  sleepers,  and  fitted  to  the  curved  side  of  the  rail, 
as  shown  at  d.  This  plank  is  on  the  outside  of  the  track.  On 
the  inner  side  the  rails  are  spiked  in  the  usual  way  with  hook- 
headed  spikes  s  s  s,  of  which  those  at  the  joint,  r,  pass  through 
a  flat  wrought-iron  plate,  P,  which  gives  a  better  bearing  to  the 
end  of  the  rail,  and  prevents  dislocation  of  parts.  Each  plank, 
as  D,  is  bolted  to  the  rail  by  four  horizontal  half-inch  bolts, 
b,  b,  b,  b,  furnished  with  nuts  and  washers  on  the  further  side  of 
D  (not  seen). 


IROX  CONSTRUCTIONS.  152 

A  modification  of  the  above  construction  consists  in  substi- 
tuting for  the  plate  P,  a  short  piece  or  strap  of  iron  fitted  to  the 
surface  of  the  inside  of  the  rail,  and  through  which  the  two 
bolts  hh,  next  to  the  joint,  pass. 

With  the  now  extended  use  of  steel  rails,  the  fish-joint,  also  in 
very  general  use,  consists  of  an  iron  fish-plate  on  each  side  of  the, 
rail,  with  two  bolts  on  each  side  of  the  joint.  This  makes  a  very- 
firm  joint.  The  plan  has  also  been  sometimes  adopted  of  having 
the  track  break  joints.  That  is,  a  joint,  as  a,  Pig.  132,  on  one  rail 
of  a  track,  is  placed  opposite  the  centre  of  the  rail  he  of  the  other 
line  of  the  same  track.  As  a  track  always  tends  to  settle  at  the 
joints,  a  jumping  motion  is  induced  in  a  passing  train,  which 
perhaps  may  be  thought  to  be  less  violent  if  only  on  one  rail  at  a 
time. 

256.  Graphical  Construction. — Tlireo  lines  through  X,  making 
angles  of  60°  witli  eacli  otlier,  will  be  llic  isometric  axes.  Reincni 
bering  that  it  is  the  relative  position  of  the  lines  which  distinguishes 
an  isometrical  drawing,  we  can  place  XX'  parallel  to  the  lower 
border,  and  thus  fill  out  the  plate  to  better  advantage.  The  rail 
being  4"  vdde  at  bottom,  and  4"  high,  circumscribe  it  by  a  square 
"Kcan,  from  the  sides  of  which,  or  from  its  vertical  centre  line,  lay 
off,  on  isometric  lines,  the  distances  to  the  various  points  on  the  rail. 
Thus,  let  the  widest  part  of  the  rail,  near  the  top,  be  3"  across, 
and  h  an  inch  below  the  top  ac.  Let  the  width  at  the  top  be  2", 
and  at  the  narrowest  part  1";  and  let  the  mean  thickness  of  thf 
lower  flange  be  f ".  The  sides  of  the  rail  are  represented  by  the 
bottom  lines  at  XX',  and  the  tangents  each  side  of  R,  to  the  curvea 
of  the  section.  Let  the  plank  D  be  6"  wide,  and  4"  high.  All 
the  lines  of  the  spikes,  ss,  are  isometrical  lines  except  their  top 
edges,  as  st.  The  curve  at  the  joint  r,  and  at  X',  are  similar  to  the 
corresponding  parts  of  the  section  at  X. 

To  secure  case  of  graphical  construction,  let  the  bolt  heads,  A, 
fee  ,  be  placed  so  that  their  edges  shall  be  isometric  lines. 

Fig.  134,  is  a  plan  and  end  elevation  of  a  heavy  cast-iron 
chair  designed  as  a  partial  equivalent  for  a  continuous  rail,  by 
making  the  outside  of  the  chair  extend  to  the  top  of  the  rail. 
The  fault  in  every  such  contrivance,  the  best  of  which  at  present 
seems  to  be  the  fish-joint,  is  that,  as  the  joint  cannot  be  made  as 
solid  as  the  unbroken  rail,  the  wave  of  depression  just  in  advance 
of  the  engine  is  more  or  less  completely  broken  at  every  joint. 

258.  Ex.  7°.  The  Hydraulic  Ram.  In  order  to  give  an  iron 
construction,  from  the  department  of  machinery,  so  as  to  render 
this  volume  a  more  fit  elementary  course  for  the  machinist  as  well 


154  IRON    COXSTKUCnONfe. 

as  for  the  civil  engineer,  a  simple  and  generally  useful  structure 
viz.  a  hydraulic  ram,  has  been  chosen,  as  a  fit  example  for  the 
last  to  bo  described  in  detail. 

This  machine  is  designed  to  employ  the  power  of  running  watei 
to  elevate  water  to  any  desired  heiglit. 

PI.  XXIII.,  Figs.  135-137,  shows  a  hydraulic  ram,  of  highly 
appioved  construction,  and  of  half  the  full  size. 

259.  Mechanical  conUructlon. — FF — F'F'  are  feet  to  support 
the  machine.  These  are  screwed  to  a  floor  or  other  firm  support. 
Ar> — A'B'B'  is  theinlet  pipe,  openinginto  the  air  chamber  C,  at  a — a!V 
and  ending  at  dd — d' d' — d"d'  the  opening  in  the  top  of  the  waste 
valve  chamber,  E — E' — E".  At  a — ah'  is  the  opening  as  just 
noticed  from  the  inlet  pipe  into  the  air  chamber  C  (not  seen  in  the 
plan).  This  opening  is  controlled  by  a  leather  valve  ee',  weighted 
A\  ith  a  bit  of  copper  e''e"\  and  is  fastened  by  a  screw  A'A'",  and 
an  oblong  washer  g'g.  At  N  and  H  arc  the  extremities  of  two 
outlet  pipes  leading  from  the  air  chamber  at  F"F"'.  Either  one, 
but  not  both  of  these  outlet  pipes  together,  may  be  used,  as  one 
of  the  exchangeable  flanges,  H'  is  solid,  while  the  otlier  is  per- 
forated, as  seen  at  M',  Fig.  137.  The  air  chamber  is  secured  by 
bolts  passing  through  its  flange  f'f\  through  the  pasteboard  or 
leather  packing,  2^P — p'l  ^"cl  the  flange  D — D'D'  at  cc.  Tliis 
flange,  and  part  of  the  inlet  pipe  are  shown  as  broken  in  the 
elevation,  so  as  to  exi)ose  the  valve  ee\  and  the  adjacent  parts. 
LL'  is  a  flange  through  which  the  inlet  pipe  passes,  and  this  pipe 
is  slit  and  bent  over  the  inner  edge  of  the  aperture  in  LL',  forming 
a  flange,  which  presses  against  a  leather  packing,  U\  and  makes  a 
tight  joint.  The  outlet  pipes  are  secured  in  tlie  same  way.  At 
uu — u'  are  the  square  heads  of  bolts  which  fasten  the  flanges  to 
the  projections  UU — U'.  K — K'  is  a  shelf  bearing  the  waste 
valve  chamber,  E — E'E",  and  the  adjacent  parts.  W — W  is  the 
flange  of  this  valve  chamber,  secui-ed  by  two  bolts  at  r^v" — v\ 
which  ])ass  through  the  leather  packing  y.  Ji'h"  is  the  waste 
valve,  perforated  with  holes,  ic,  to  allow  water  to  flow  through  it. 
mm'  is  the  valve  stem,  d'd'k'k'  is  a  perforated  standard  serving 
as  a  guide  to  the  valve  stem,  and  also  as  a  support  to  the  hollow 
screw  s.  w  is  a  rest,  secured  to  the  valve  stem  by  a  pin  p".  ^' 
is  a  nut,  part  of  which,  qq'^  is  made  hexagonal,  r  is  a  "jam'' 
nut  (253). 

In  the  plan  of  this  portion  of  the  machine,  the  innermost  circle 
IH  the  top  of  the  valve  stem;  next  is  the  body  of  the  valve  stem  ; 
next,  the  top  of  the  rest;  next,  the  bottom  of  the  same;  next,  the 


:pi..xxii 


c 


c 


inOX   COXSTRtJCTlOXS.  J  55 

nut  q"  ;  and  outside  of  that,  and  resting  on  the  toj)  of  tlie  waste 
valve  chamber,  are  the  standards,  dd. 

960.  Operation. — Principles  ifivolved. — In  the  case  of  wliat 
miglit  l)c  called  passive  consti'UctionSy  that  is  mere  stationary  sup- 
ports, like  bridges,  &c.,  a  knowledge  of  the  construction  of  tho 
parts  enables  one  to  proceed  intelligently  in  making  a  drawing ; 
hut,  in  the  case  of  what  may,  in  opposition  to  the  foregoing,  be 
called  active  constructions^  or  machines,  a  knowledge  of  their  mode 
of  operation  is  usually  essential  to  the  most  expeditious  and  accu- 
rate graphical  construction  of  them,  because  a  machine  consists 
of  a  train  of  connected  pieces,  so  that  a  given  position  of  any  piece 
implies  a  corresponding  position  for  every  other  pirt.  Having, 
then,  in  a  drawing,  assumed  a  definite  position  for  some  important 
part,  the  remaining  parts  must  be  located  from  a  knowledge  of  the 
machine,  though  drawn  by  measurements  of  the  dimensions  of  that 
part.  OnXj fixed  bearings^  and  centres  of  niotio7X,  can  properly  be 
located  by  measurement,  in  machine  drawing. 

The  principles  involved  in  the  operation  of  the  hydraulic  ram 
may  be  summed  up  under  three  heads,  as  follows: 

261.  I.  Work.  a.  When  a  certain  iceight  is  moved  through  a 
certain  space^  a  certain  amount  of  xoorh  is  expended. 

h.  Thus ;  when  a  quantity  of  water  descends  through  a  certain 
space,  a  certain  amoimt  of  work  is  developed. 

c.  As  the  idea  of  work  involves  the  idea  both  of  weight  moved, 
and  space  traversed,  it  follows  that  tcorks  may  be  equal,  while  the 
weights  and  spaces  may  be  unequal.  Thus  the  work  developed  by 
a  certain  quantity  of  water,  while  descending  through  a  certain 
height,  may  be  equal  to  that  expended  in  raising  a  portion  of  that 
water  to  a  greater  height. 

262.  II.  Equilibrium,  a.  Where  forces  are  balanced,  or  mutu- 
ally neutralized,  they  are  said  to  be  in  equilibrium.  Now  the  usual 
fact  is,  that  when  such  equilibrium  is  disturbed,  it  does  not  restore 
Itself  at  once,  but  gradually,  by  a  scries  of  alternations  about  the 
ftate  of  equilibrium.  Thus  a  stationary  pendulum,  being  swung 
from  its  position  of  equilibrium,  does  not,  at  the  first  returning 
vibration,  stop  at  the  lowest  point,  but  does  so  only  after  many 
vibrations. 

b.  Theoretically,  these  vibrations,  as  in  the  case  of  the  pendulum, 
would  never  stop,  but  in  practice  the  resistance  of  the  air,  fription, 
&c.,  make  a  continual  supply  of  a  greater  or  less  amount  of  forc^ 
necessary  to  perpetuate  the  alternations  about  the  position  oi 
state  of  equilibrium. 


3.56  rnox  constructions. 

263.  m.  A  physical  fact  taken  account  of  in  the  hydraulic  ram, 
18,  that  water  in  contact  with  compressed  air  will  absorb  a  certain 
portion  of  such  air. 

264.  Passing  now  more  particularly  to  a  description  of  the  ope 
ration  of  the  hydraulic  ram:  1°.  Water  from  some  elevated  pond 
or  reservoir  flows  into  the  machine,  through  the  inlet  pipe  AA' 
and  continues  through  the  machine,  and  flows  out  through  the  hole 
in  the  waste  valve  A'A",  pressing  meanwhile  against  the  solid  parts 
of  the  roof  of  this  valve,  whose  hollow  form  —  open  at  the  bottonj 
—  is  clearly  shown  in  Fig,  136. 

2°.  Presently  the  water  acquires  such  a  velocity  as  to  press  so 
strongly  against  the  roof  of  the  Avaste  valve,  that  this  valve  is  lifted 
against  the  under  side  of  the  roof  of  its  chamber  which  it  fita 
accurately. 

3*^.  The  water  thus  instantly  checked,  expends  its  acquired  force 
in  rushing  through  the  valve  e — e'e'"  and  in  compressing  the  air  in 
*.he  air  chamber  C. 

4°.  The  holes  F"  or  F'"  of  the  outlet  pipe,  leading  to  an  unob- 
structed outlet,  the  compressed  air  immediately  forces  the  water 
out  through  the  outlet  pipe  until,  after  a  number  of  repetitions  of 
this  chain  of  operations,  the  portion  of  the  water  thus  expelled 
from  the  air  chamber  is  i-aised  to  a  considerable  height. 

5°.  In  accordance  with  the  second  principle,  the  flow  of  water  from 
the  air  chamber  does  not  cease  at  the  moment  when  the  confined 
air  is  restored  to  its  natural  density,  but  continues,  so  that — taking 
account  also  of  the  absorption  of  the  air  by  the  water  at  the  time 
of  compression — for  a  moment  the  air  of  the  air  chamber  is  moi-e  rare 
than  the  external  atmosphere.  Hence  to  keep  a  constant  supply 
of  air  to  the  air  chamber,  a  fine  hole  called  a  snifting  hole,  is  punc- 
tured, as  with  a  needle,  at  ss\  i.e.,  just  at  the  entrance  of  the  inlet 
pi]ie  into  the  machine.  Through  this  hole  air  enters,  with  a  snift- 
ing sound,  when  the  flow  of  water  recommences,  so  as  to  supply 
the  air  chamber  with  a  constant  quantity  of  air.  When  the  waste 
valve  is  at  the  bottom  of  the  chamber  EE',  the  nut  and  "jam" 
are  together  at  the  bottom  of  the  screw  s\  and  the  valve  is  at 
liberty  to  make  a  full  stroke.  By  raising  the  valve  to  its  highest 
point  and  turning  the  nut  and  "jam''  to  some  position  as  shown 
in  the  figure,  the  stroke  of  the  valve  can  be  shortened  at  pleasure, 
and,  at  its  lowest  point,  will  be  as  far  from  the  bottom  of  the 
chamber  as  the  "jam,"  q'\  is  above  its  lowest  position. 

266.  In  practice,  it  is  found  that  the  strokes  of  the  waste  valvfl 
ihortly  become  regular ;  their  frequency  depending  in  any  given 


IRON    CONSTRUCTIONS.  157 

case  on  the  height  of  the  supply  reservoir,  the  height  of  the  ejected 
column,  the  size  of  the  machine,  the  length  of  the  stroke  of  the 
valve,  &c. 

267.  The  proportion  of  water  discharged  into  the  receiving 
eservoir  will  also  depend  on  the  above  named  circumstances, 
oeing  more  or  less  than  one  third  of  the  quantity  entering  the 
machine  at  AA'.  In  a  machine  by  M.  Montgolfier  of  France,  said 
to  be  the  original  inventor — water  falling  7tV  feet,  raised  ^j  of  itself 
to  a  height  of  50  feet. 

2G8.  Graphical  Construction. — Scale;  half  the  full  size.  u.  Hav- 
ing the  extreme  dimensions  ol  the  plan,  in  round  numbers  9"  and 
12",  proceed  to  arrange  the  ground  line,  leaving  room  for  the  plan 
below  it. 

b.  Draw  a  centre  line,  NC,  for  plan  and  elevation,  about  in  the 
middle  of  the  width  of  the  plate. 

c.  Draw  a  centre  line,  AK,  for  the  plan,  parallel  to  the  ground  line. 

d.  Exactly  4^"  from  the  centre  line  NC,  draw  the  centre  line  vu" 
— K'm'  for  the  waste  valve  chamber  and  parts  adjacent. 

e.  With  the  intersection,  *,  of  the  centre  lines  of  the  plan,  as  a 
centre,  draw  circles  having  radii  of  \%"  and  3Jj"  respectively,  and 
through  the  same  centre,  draw  diagonals,  as  cc. 

f.  On  the  centre  line,  NC,  are  the  centres  of  the  circles,  F"F"', 
'a'hose  circumferences  come  within  p'^  of  an  inch  of  the  inner  onw 
of  the  two  circles  just  drawn. 

g.  Draw  the  valve,  e,  the  copper  weight  e",  the  screw  end,  A, 
and  the  nut  and  oblong  washer,  It,"  and  g. 

h.  Locate,  at  once,  the  centres  of  all  the  small  circles,  cc,  cfec,  by 
the  intersection  of  arcs,  \"  from  the  circle /»jo  having  *  for  a  centre, 
with  the  diagonals ;  then  proceed  to  draw  tliese  circles. 

i.  Draw  the  projections,  as  U,  drawing  the  opposite  ones  simul- 
taneously, and  using  an  auxiliary  end  view  of  the  nuts  m,  as  ollen 
explained  before. 

J.  Draw  the  feet,  F,  with  their  grooves,  F,  and  bevel  edged  screw 
holes,  L". 

k.  In  drawing  the  shelf,  K,  and  flange  W,  the  intersection  of  the 
centre  lines  BK  and  m'm,  is  the  centre  for  the  curves  which  inter- 
sect the  centre  line  AK;  the  corners,  1  1,  of  the  nuts,  v,v'\  are  the 
centres  for  the  curves  that  cross  the  centre  line,  vv" ;  and  the 
remaining  outlines  of  the  shelf  are  tangents  to  the  arcs  thus  drawn, 
and  those  of  the  flange  are  lines  sketched  in  so  as  to  give  curves 
tangent  to  the  arcs  already  drawn,  and  short  straight  lines  parallel 
to  w" 


158  IRON    COXSTRUCTIONS. 

/,  The  reiiKiining  cii-cles  and  larger  hexagon,  u\  of  this  portiorj 
of  the  plan,  have  the  intersection  of  the  centre  lines  for  a  centre ;  and 
inav  be  drawn  by  nieasuiements  independently  of  the  elevation,  or 
by  projection  from  the  elevation,  after  that  shall  have  been 
finished. 

269.  Passing  to  the  elevation; — 

a.  Constrnct,  at  one  position  of  the  T  square,  the  horizontal  lines 
of  both  feet;  then  the  horizontal  lines  of  the  nuts  w',  and  flange  L', 
and  projection  U' ;  with  the  horizontal  lines  of  the  floor  of  the  air 
chamber  and  adjacent  parts. 

h.  Project  up  from  the  plan  the  vertical  edges  of  the  feet,  F'F', 
the  flange,  nut,  and  projection  L',  u'  and  TJ',  the  valv?  e\  the  cop- 
per e",  the  screw  A",  the  washer  ^,  the  air  chamber  flange/"/'',  and 
screw  z.  Break  away  the  portion  D — see  plan — of  the  body  of 
the  machine,  and  the  near  wall  of  the  water  channel  A'B',  Break 
away  also  the  further  wall  of  the  water  channel  so  as  to  show  3 
Bection,  ir,  of  the  further  outlet  pipe,  H — see  plan.  Q  's  the  centre 
of  the  spherical  part  of  the  air  chamber  to  which  the  conical  part 
is  tangent. 

c.  Draw  all  the  horizontal  lines  of  the  waste  valve  chamber  and 
parts  adjacent.  Make  the  edges  of  the  threads  of  the  screw  straight 
ami  slightly  inclined  upwards  toward  the  right. 

d.  Project  up  from  the  plan,  or  lay  ofl",  by  measurement,  the 
widths  of  various  parts  through  which  the  valve  stem  passes,  and 
draw  their  vertical  edges. 

Fig.  136  is  a  section  of  the  waste  valve  chamber,  showing  part 
both  of  the  interior  and  exterior  of  the  waste  valve.  The  dotted 
circles  form  an  auxiliary  plan  of  this  valve,  in  which  the  holes  have 
two  radial  sides,  and  two  circular  sides  with  x"  as  a  centre.  The 
top  of  the  valve  is  conical,  so  that  in  the  detail  below,  two  of  the 
Bides  of  the  hole  /i,  tend  towards  the  vertex,  x.  At  ii'^  one  of  these 
holes,  of  which  there  are  supposed  to  be  five,  is  shown  in  section. 

Fig.  137.  The  outlines  of  M,  one  of  the  outlet  i)i])e  flanges,  are 
drawn  by  processes  similar  to  those  employed  in  drawing  the  shelf, 
K,  in  plan. 

270.  Kxecution.  As  a  line  drawing,  the  plate  explains  itself.  It 
would  make  a  very  beautiful  shaded  drawing  and  one  that  the 
oarelul  student  of  the  chapter  on  shading  and  shadows,  would  be 
able  to  execute  with  substantial  accuracy,  without  further  izi3truo 
Lion. 

We  conclude  this  division  with  the  following  additional  excr- 
•isee  as  examples  of  iron  constructions — one  from  civil  engineer- 


IRON   CONSTRUCTIONS. 


159 


ing  practice,  the  other  three  from  mechanical  engineering;  styl- 
ing them  exercises,  since,  being  partially  shown  (yet,  with  the 
description,  sufficiently  so  for  their  purpose),  they  leave  some- 
thing to  be  supplied  by  the  student  from  the  general  insight 
gained  from  previous  practice. 

Exercise  1.  A  Stop-valve. — The  following  figure  shows  one  of  many 
forms  of  valve  differing  more  or  less  in  detail,  and  made  for  the  purpose 
of  shutting  off  the  passage  of  steam,  water,  etc.,  through  pipes.     Such 


halves  either  lift  'rom  their  seats,  as  in  the  example  shown,  or  slide  off 
them,  in  which  case  they  are  sometimes  called  gates. 

The  figure  represents  what  is  called  a  ghhc-vahe,  from  the  general 
external  form  of  its  valve-chamber  NCCL.  In  this  chamber  is  a  bent  par- 
tition, or  diaphram,  CEC,  containing  the  seat,  E,  of  the  valve  D.  This 
valve  is  raised  or  lowered  by  means  of  the  hand-wheel  K,  and  screw 
valve-stem  F  working  in  the  collar  G,  which  is  screwed  into  the  top, 
NC,  of  the  chamber. 

The  head  at  the  bottom  of  the  valve-stem,  working  loosely  in  the  hol- 
low head  of  the  valve,  raises  tl>e  latter  vertically  without  turning  it. 
The  cap  H  secures  the  necessary  packing.  Opening  the  valve  then 
allows  of  the  passage  of  any  fluid  through  AB  and  the  pipes  which  may 


160  mO'S    CONSTRUCTIONS. 

be  attached  at  A  and  B.  Thesn  openings  are  from  J"  to  2"  diameter. 
The  measurements  and  scale  may  thererore  be  assumed,  and  plan  and 
end  elevation  added. 

Exercise  2.  An  iron  truss  bridge.  Pi  XXIV.,  Figs.  1-7. — This  bridge 
is  partly  of  wrought,  and  partly  of  cast  iron,  and  known  from  its  form 
and  its  inventor  as  Whipple's  trapezoidal-trtiss  bridge. 

The  ujiper  chords,  a — a'a\  are  lioUow  cast-iron  cylinders  7^"  diameter, 
and  I"  to  %"  thickness  of  metal.  The  j)o^ts,  p',  and  struts,  3,  S'S",  arq 
also  of  cast-iron,  the  latter,  double,  as  seen  in  the  fragment  of  end  ele- 
vation. Fig.  2,  and  fragment  of  plan.  Fig.  3. 

The  posts  extend  through  the  flooring,  where  they  are  5"  in  diameter, 
and  rest  on  seats  on  the  tops  of  the  cast-iron  coupling  blocks,  n'n',  as 
shown  in  the  plan.  Fig.  5,  and  end  elevation,  Fig.  6,  of  one  of  these 
blocks. 

The  loicer  chord,  bl> — b'V,  is  composed  of  heavy  wrought-iron  rods  made 
in  links  embracing  two  successive  coupling-blocks,  in  the  manner  shown 
in  Figs.  4-6.  The  two  end  lengths,  however,  are  single,  as  shown, 
and  are  secured  by  nuts,  g',  at  the  outer  end  of  the  shoes  s,  s",  which 
holds  the  feet  of  the  struts  SS'S". 

The  structure  is  further  held  in  shape,  and  the  forces  acting  in  it  suit- 
ably sustained  and  distributed  by  the  diagonal  and  vertical  rods  r'r', 
each  of  which,  after  the  first  two  from  the  end,  crosses  two  panels  of  the 
bridge,  as  the  spaces  between  the  posts  are  called. 

The  horizontal  diagonal  rods,  y,  under  the  floor,  tightened  by  links  I 
woiking  on  right-  and  left-handed  screws  (Div.  V.,  Ex.  9)  in  the  adja- 
cent rod  ends,  provide  against  the  horizontal  force  of  winds.  The  light 
transverse  flanged  beams  k — k",  overhead,  also  help  to  stiffen  th6  struc- 
ture laterally. 

The  main  transverse  beams  c — c' — c"  rest  on  the  coupling-blocks,  and 
support  the  floor  joists  dd'd",  on  which  the  floor  planks  gg'g"  rest. 
CCC"  is  the  coping,  serving  to  cover  the  irregular  ends  of  the  floor 
planks,  and  as  a  guard  to  prevent  vehicles  from  striking  the  truss. 

Fig.  7  is  an  enlarged  view  of  the  centre  joint  where  the  two  halves  of 
the  posts  meet. 

Other  useful  details  would  be  vertical  longitudinal  sections  of  the 
joints  as  ee"  and  e',  which  Avould  show  an  opening  in  the  under  side  of 
the  upper  chord,  sufficient  for  the  entrance  of  the  diagonal  rods,  and 
tliese  rods  forged  into  rings  clasping  the  stout  wroiiglit-iron  pins  e,  e'c" \ 
also  the  level  bearing  for  the  head  of  the  post,  except  at  the  joint  at  the 
head  of  the  strut. 

Tlie  three  top  cross-beams  indicated  at  Teh,  Fig.  3,  show  that  a  slcew- 
bridge  is  represented,  that  is,  one  which  crosses  the  stream  obliquely, 
the  extreme  timber,  h,  being  parallel  to  the  length  of  the  stream. 


jL'j^JCxar 


IROX    COXSTRUCTIOXS.  161 

The  span  of  the  bridge  is  114  leet,  in  12  equal  panels  of  9^  feet  eacn; 
the  roadway  is  19  feet  wide  from  centre  to  centre  of  the  trusses,  which 
are  15  feet  9  inches  in  height  from  the  centre  of  the  coupling-blocks,  n', 
to  that  of  the  upper-chord  pins  as  at  e'. 

Suitable  scales  are  3  to  5  feet  to  1  inch  for  the  general  views,  and 
from  6  inches  to  1  foot  to  one  inch  for  tht;  details. 

Exercise  3.  A  vertical  loiler.  PI.  XXIV.,  Fig.  8. — This  figure,  being 
given  partly  as  an  excellent  example  in  shading,  and  of  certain  flame 
effects  instructive  to  the  draftsman,  is  described  without  letters  of  ref- 
erence. 

The  figure  represents  a  vertical  section  of  what  is  known  as  the  Shap- 
ley  patent  boiler,  differing  from  the  ordinary  tubular  vertical  boiler  as 
appears  from  the  figure  and  following  description. 

The  central  combustion  chamber,  being  tall,  is  designed  to  effect 
three  results ;  viz.,  to  raise  its  top,  called  tlie  crown  sheet,  so  far  above 
the  fire  as  to  retard  burning  out;  to  afford  abundant  room  for  perfect 
combustion,  thereby  generating  more  heat;  and  to  effectually  convey 
this  heat  to  the  water  which  surrounds  the  fire-box  in  a  thin  sheet. 

Heat  is  further  conveyed  to  the  water  by  passing,  as  shown  by  the 
arrows,  through  short  transverse  tubes,  two  of  them  seen  in  section,  and 
vertical  tubes  between  the  fire-box  and  the  outer  shell.  These  open  into 
the  annular  base  flue  (interrupted  by  the  ash-pit  door),  which  leads  to 
the  smoke-pipe  (sometimes  called  the  uptake). 

The  upper  section,  or  steam-dome,  is  mostly  occupied  by  steam,  and 
is  stayed  by  bolts  to  the  crown  sheet. 

Since  the  tubes,  when  sooty,  lose  much  of  their  heat-conducting  power, 
they  are,  in  this  boiler,  made  very  easily  accessible  for  frequent  clean- 
ing by  connecting  the  two  sections  of  the  boiler  by  a  double  annular 
jacket  which  contains  no  steam  or  water  and  sustains  no  pressure.  It  is 
made  in  sections  for  easy  removal,  and  thus  allows  ready  access  to  the 
tubes. 

Exercise  4.  A  direct- acting/  steam-pump.  PI.  XXIV.,  Fig.  9. — The  mag- 
nitude and  variety  of  pumping  requirements  for  water,  oil,  and  various 
other  liquids,  hot  or  cold,  thin  or  viscid,  pure  or  gritty,  and  for  drain- 
age, mining,  city,  hotel,  railroad  -  station,  and  other  purposes,  have 
called  forth  a  large  amount  of  inventive  talent  and  many  ingenious  and 
effective  pumping  engines. 

The  figure  represents  a  vertical  longitudinal  section  of  the  Knowles 
steam-pump,  affording  a  useful  study  and  guide  in  making  a  finished 
drawing. 

BB  is  the  pump  barrel  or  water  cylinder — lined,  when  the  character 
of  the  fluid  to  be  pumped  requires  it,  with  composition  linings,  shown 
at  XX  and  similarly  in  section  on  the  upper  side  of  the  barrel.     P  is  the 


IQ2  IROIS"  COXSTRUCTIOIirS. 

•water  piston  with  its  packing  p,  and  secured  to  the  piston-rod  a  by 
nut  and  lock-nut  (Div.  V.,  12)  seen  at  the  left. 

JEF  is  the  steam  cylinder  with  its  piston  on  the  same  piston-rod,  a, 
with  the  water  piston,  thus  forming  what  is  called  a  direct-acting  pump. 
Both  cylinders  are  provided  with  stuffing-boxes  hg  and  K. 

The  pump  valves  under  the  letter  b  are  here  shown  as  lifting  disk 
valves  circular  in  plan,  but  may  be  cage,  or  hinge,  or  any  other  valves. 

In  the  posiiion  shown,  and  the  piston  still  moving  towards  the  left, 
water  is  entering  through  the  lower  or  suction  circular  inlet  and  the 
lower  right-hand  valve,  and  is  discharging  by  the  upper  or  discharge 
pipe,  which  is  smaller  than  the  suction  pipe.  By  raising  the  upper  left- 
hand  valve  the  discharge  water  also  partly  enters  the  air-chamber  A, 
where,  by  compressing  the  confined  air,  a  steady  discharge  is  obtained. 
The  valves  rise  and  fall,  each  working  on  a  short  spindle,  and  are 
quickly  closed  by  the  aid  of  sjiiral  springs  above  them;  seen  on  the  two 
closed  valves. 

The  steam  and  exhaust  ports  and  passages  to  the  steam  cylinder  are 
of  the  usual  form  ;  n,  the  orifice  for  the  admission  of  steam  from  the 
boiler,  and  the  central  orifice  is  the  exhaust.  The  steam-valve  is  a 
double  D  valve. 

A  stroke  to  the  right  being  about  to  begin,  a  roller  on  the  opposite 
side  of  the  tappet  arm  CC,  carried  by  the  piston-rod  a,  raises  the  left 
end  of  the  rocker  DD.  This,  by  means  of  the  link  s,  slightly  rotates  the 
valve-rod  Z  and  its  "chest-piston,"  Fig.  11,  so  as  to  bring  it  into  a  posi- 
tion to  take  steam  through  the  small  passage  at  the  lower  right-hand 
corner  of  the  steam-chest  G,  which  throws  the  piston  to  the  opposite  end 
of  its  stroke,  carrying  the  valve  by  means  of  its  stem  T,  Fig.  10.  Steam 
can  then  enter  the  left-hand  end  of  the  cylinder  through  the  left-hand 
chamber  of  the  D  valve,  while  exhaust  steam  escapes  through  the  pas- 
sage y  and  the  right-hand  chamber  into  the  central,  or  exhaust  passage. 

At  i  is  the  valve-rod  guide.  j\a  a  collar  on  the  valve-rod.  u  clamps 
the  rocker  connection  to  the  valve-rod.  t  adjusts  the  link  s.  M  is  the 
oil-cup,  and  7i  a  stud  to  attach  a  hand  lever. 

These  pumps  are  made  of  a  large  range  of  sizes,  from  water  cylinders 
of  2",  and  steam  cylinders  of  3J"  diameter,  and  4"  stroke;  to  water 
cylinders  of  20",  and  steam  cylinders  of  28"  diameter,  and  12"  stroke. 
Fig.  9  may  be  regarded  for  drawing  purposes  as  a  sketcii  (from  a  scale 
drawing  and  in  true  proportion,  however)  of  a  pump  having  a  water 
cylinder  of  7",  and  a  steam  cylinder  of  12"  diameter,  with  a  12"  stroke. 

For  variety  of  practice  in  the  use  of  scales,  the  pump  may  then  be 
drawn  on  a  scale  of  i  or  \  the  full  size,  with  details  on  scales  of  3"  to  a 
foot,  or  of  full  size. 

THE   END. 


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